The Classification of Three-Dimensional Homogeneous Complex Manifolds
This book provides a classification of all three-dimensional complex manifolds for which there exists a transitive action (by biholomorphic transformations) of a real Lie group. This means two homogeneous complex manifolds are considered equivalent if the
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
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Jorg Winkelmann
The Classification of Three-dimensional Homogeneous Complex Manifolds
Springer
Author Jorg Winkelmann Ruhr-Universitat Bochum Mathematisches Institut NA 4 0-44780 Bochum, Germany E-mail: [email protected]
Mathematics Subject Classification (1991): 32MIO, 20G20, 22EI 0, 22E15, 32L05, 32M05
ISBN 3-540-59072-2 Springer-Verlag Berlin Heidelberg New York
CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready output by the author SPIN: 10130297 46/3142-543210 - Printed on acid-free paper
Preface
In this monograph we give a classification of all three-dimensional homogeneous complex manifolds. A complex manifold X is called homogeneous if there exists a connected complex or real Lie group G acting transitively on X as a group of biholomorphic transformations. The goal is to cl&88ify these manifolds up to biholomorphic equivalence. Since the cl&88 of homogeneous complex manifolds is much too big for any serious attempt of complete classification, it is necessary to impose further conditions. For example E. Cartan classified in [Cal symmetric homogeneous domains in en. Here we will require that X is of small dimension. For dimc(X) = 1 the classification follows from the Uniformization Theorem. In 1962 J. Tits classified the compact homogeneous complex manifolds in dimension two and three [Til]. In 1979 J. Snow classified all homogeneous manifolds X = G/ H with dimc(X) 3, G being a solvable complex Lie group and H discrete [SJ1]. The classification of all complex-homogeneous (i.e. G is a complex Lie group) twodimensional manifolds was completed in 1981 by A. Huckleberry and E. Livorni [HL]. Next, in 1984 K. OeJjeklaus and W. Richthofer claesified all those homogeneous two-dimensional complex manifolds X = G/ H where G is only a real Lie group [OR]. The classification of three-dimensional complex-homogeneous manifolds was completed in 1985 [WI]. Finally in 1987 the general classificetion of the three-dimensional homogeneous complex manifolds was given by in [W2]. The purpose of this monograph is to give the complete proof of the classification of three-dimensional complex manifolds G/ H . I would like to use this opportunity to thank Alan Huckleberry for his support and encouragement. I would also like to thank Wilhelm Kaup, Karl OeJjeklaus and Eberhard Oelje
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