Homogeneous complex surfaces
type: Item
instance of: scholarly article
title: Homogeneous complex surfaces
zbMATH Open document ID: 0551.32023
DOI: 10.1007/BF01457059
author: Karl Oeljeklaus
author: Wolfgang Richthofer
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182920
review text: The content of this paper is a classification of all two dimensional complex manifolds which are homogeneous under the action of a real Lie group of holomorphic automorphisms. The problem of classifying homogeneous surfaces has been considered in various different settings and of these results we note the following. Compact homogeneous surfaces were classified by \textit{J. Tits} [Comment. Math. Helv. 37, 111-120 (1962; Zbl 0108.363)], all complex homogeneous surfaces (a complex Lie group acts holomorphically) were given by \textit{A. T. Huckleberry} and \textit{E. L. Livorni} [Can. J. Math. 33, 1097-1110 (1981; Zbl 0504.32025)] and the list of pseudo-concave homogeneous surfaces is contained in the work of \textit{A. T. Huckleberry} and \textit{D. Snow} [Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV. Ser. 8, 231-255 (1981; Zbl 0464.32019)]. Assuming \(X=G/H\), where X is neither complex-homogeneous nor a product of homogeneous Riemann surfaces, we use the \({\mathfrak G}\)-anticanonical fibration of \textit{A. Huckleberry} and \textit{E. Oeljeklaus} [''Classification theorems for almost homogeneous spaces'', Inst. Elie Cartan, Univ. Nancy I 9 (1984; Zbl 0549.32024)] to show that X is a covering of \(Y:=G/I,\) which is an open submanifold of \(\hat Y:=\hat G/\hat I,\) a complex-homogeneous surface equivariant (but not necessarily closed) in some \({\mathbb{P}}_ N\). Further the algebraic group \(\hat G'\) acts transitively on \(\hat Y,\) so \(\hat Y\) is Zariski open in its Zariski closure. If no solvable group acts transitively on X, then X is either pseudoconcave and the group acting is semisimple, or it is \({\mathbb{C}}^ 2\setminus {\mathbb{R}}^ 2\) or one of its coverings and the group is a semi- direct product of \({\mathbb{R}}^ 2\) with \(SL_ 2({\mathbb{R}}^ 2)\). If the group acting is solvable, then \(\hat Y={\mathbb{C}}^ 2\) and the problem is to analyse fibrations G/I\(\to G/J\) with one dimensional fiber and base. The fact that these are real analytic but not h…
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 3877544
zbMATH Keywords: homogeneous two dimensional complex manifolds
zbMATH Keywords: action of real Lie group of holomorphic automorphisms
zbMATH Keywords: CR-structure
zbMATH Keywords: Heisenberg group
MaRDI profile type: Publication
cites work: Q3895790
cites work: Sur les domaines bornes homogenes de l'espace de \(n\) variables. complexes
cites work: Q5580320
cites work: Analytische Faserungen über holomorph-vollständigen Räumen
cites work: Q5542941
cites work: Q4102494
cites work: A Classification of Homogeneous Surfaces
cites work: Q3342885
cites work: Q3915257
cites work: Q4089001
cites work: Q5658761
cites work: Q5565587
cites work: On almost homogeneous compact complex analytic surfaces
cites work: On the pseudo-conformal geometry of hypersurface of the space of \(n\) complex variables
cites work: Espaces homogenes complexes compacts
cites work: Heisenberg Manifolds and Theta Functions
description: scientific article; zbMATH DE number 3877544