On proper actions of Lie groups of dimension n 2 + 1 on n-dimensional complex manifolds

• Published in: Journal of mathematical analysis and applications Vol. 342; no. 2; pp. 1160 - 1174
• Main Author: Isaev, A.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.2008
• Subjects: Complex manifolds; Proper group actions; Complex manifolds; Proper group actions
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2007.12.050

We explicitly classify all pairs ( M , G ) , where M is a connected complex manifold of dimension n ⩾ 2 and G is a connected Lie group acting properly and effectively on M by holomorphic transformations and having dimension d G satisfying n 2 + 2 ⩽ d G < n 2 + 2 n . We also consider the case d G = n 2 + 1 . In this case all actions split into three types according to the form of the linear isotropy subgroup. We give a complete explicit description of all pairs ( M , G ) for two of these types, as well as a large number of examples of actions of the third type. These results complement a theorem due to W. Kaup for the maximal group dimension n 2 + 2 n and generalize some of the author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.