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A Classification of Homogeneous Surfaces

Published online by Cambridge University Press:  20 November 2018

A. T. Huckleberry  and
E. L. Livorni
A. T. Huckleberry
Affiliation:
University of Notre Dame, Notre Dame, Indiana
E. L. Livorni
Affiliation:
University of Notre Dame, Notre Dame, Indiana
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Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1097 - 1110
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Borel, A. and Remmert, R., Uber kompakte homogène Kàhlersche Mannigfaltigkeiten, Math. Ann. 140 (1962), 429–439.Google Scholar
2. Bourbaki, , Lie groupes, Chapter 1.Google Scholar
3. Erdman-Snow, J., Solv-manifolds of dimension two and three, Notre Dame thesis.Google Scholar
4. Gilligan, B. and Huckleberry, A. T., Pseudoconcave homogeneous surfaces, Comm. Math. Helv. 53 (1978), 429–438.Google Scholar
5. Gilligan, B. and Huckleberry, A. T., On non-compact complex nil-manifolds, Math. Ann. 238 (1978), 39–49.Google Scholar
6. Huckleberry, A. T., The Levi problem on pseudoconvex manifolds which are not strongly pseudoconvex, Math. Ann. 219 (1976), 127–137.Google Scholar
7. Huckleberry, A. T. and Oeljeklaus, E., A characterization of homogeneous cones, in preparation.Google Scholar
8. Huckleberry, A. T. and Snow, D., Pseudoconcave homogeneous manifolds, in preparation.Google Scholar
9. Humphreys, J., Linear algebraic groups, Graduate texts in mathematics 21 (Springer-Verlag, 1975).Google Scholar
10. Matsushima, Y., Fibres holomorphes sur un tore complex, Nagoya Math. J. H (1958), 1–24.Google Scholar
11. Matsushima, Y., Espaces homogènes de Stein des groupes de Lie complexes I, Nagoya Math. J. 16 (1960), 205–218.Google Scholar
12. Potters, J., On almost homogeneous compact complex analytic surfaces, Invent. Math. 8 (1969), 224–266.Google Scholar
13. Tits, J., Espaces homogènes complexes compacts, Comm. Math. Helv. 37 (1962), 111–120.Google Scholar