The Classification of Three-Dimensional Homogeneous Complex Manifolds
The Classification of Three-Dimensional Homogeneous Complex Manifolds
复制标题
三维齐次复流形的分类
DOI：
10.1007/bfb0095839
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发表时间：
1995
影响因子：
1.4
通讯作者：
J. Winkelmann
中科院分区：
文献类型：
--
作者：
J. Winkelmann
如果存在连通的复李群或实李群G作为一组生物全纯变换传递作用于X，则称复流形X为齐次流形。目标是齐次复流形的一般分类。由于齐次复流形的类别太大，以至于无法进行任何认真的完全分类，因此有必要施加进一步的条件。例如E. Cartan在[Ca]对称齐次域上的分类。这里我们要求X的维数很小。对于dim (X) = 1，分类遵循均匀化定理。1962年J. Tits对二维和三维的紧齐次复流形进行了分类[Ti1]。1979年J. Snow分类了所有齐次流形X = G/H，其中暗淡的(X)≤3，G是可解复李群，H离散[SJ1]。a . Huckleberry和E. Livorni于1981年完成了所有复齐次二维流形（即G是一个复李群）的分类[HL]。接下来，在1984年，K。Oeljeklaus和W. Richthofer分类了所有齐次二维复流形X = G/H，其中G只是一个实李群[OR]。三维复齐次流形的分类于1985年完成[W1]。最后在1987年，我们的论文[W2]给出了三维齐次复流形的一般分类。本笔记的目的是描述这些流形，并简要概述分类中涉及的方法。
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 a general classification of homogeneous complex manifolds. Since the class 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 [Ca] symmetric homogeneous domains in ℂn. Here we will require that X is of small dimension. For dim ℂ(X) = 1 the classification follows from the uniformization Theorem. In 1962 J. Tits classified the compact homogeneous complex manifolds in dimension two and three [Ti1]. In 1979 J. Snow classified all homogeneous manifolds X = G/H with dim ℂ(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. Oeljeklaus and W. Richthofer classified 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 [W1]. Finally in 1987 the general classification of the three-dimensional homogeneous complex manifolds was given by our Dissertation [W2]. The purpose of this note is to describe these manifolds and briefly outline the methods involved in the classification.