On proper actions of Lie groups of dimension n2 +1 on n-dimensional complex manifolds

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 dG satisfying n2 + 2 dG < n2 + 2n. We also consider the case dG = n2 + 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 n2 + 2n and generalize some of the author’s earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group. © 2007 Elsevier Inc. All rights reserved.