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 d_G satisfying n² + 2 ≤ d_G < n² + 2 n. We also consider the case d_G = 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 + 2 n and generalize some of the author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.