Proper actions of high-dimensional groups on complex manifolds

Abstract

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. These results extend—in the complex case—the classical description of manifolds admitting proper actions of groups of sufficiently high dimensions. They also generalize some of the author’s earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.