Proper actions of high-dimensional groups on complex manifolds

A V Isaev

doi:10.1007/s13373-015-0069-7

Abstract

Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently high, all proper effective actions can be explicitly determined, and our principal goal is to provide a comprehensive exposition of known classification results in the complex setting. They include a complete description of Kobayashi-hyperbolic manifolds with high-dimensional automorphism group, which is a case of special interest.

Highlights

The purpose of this survey is to give a comprehensive overview of classification results for proper Lie group actions on smooth manifolds, with emphasis on the complexgeometric setting, where we assume that the group dimension is “large” with respect to the dimension of the manifolds
Given a Lie group G acting on a smooth manifold X by diffeomorphisms, the action is called proper if the map
One should note that the investigation of groups of motions was primarily conducted in the local setting, whereas proper group actions concern Riemannian isometry groups considered globally

Summary

Introduction

The purpose of this survey is to give a comprehensive overview of classification results for proper Lie group actions on smooth manifolds, with emphasis on the complexgeometric setting, where we assume that the group dimension is “large” with respect to the dimension of the manifolds. To the local setting, the dimension of a Lie group G acting properly and effectively on a smooth n-dimensional manifold X does not exceed n(n + 1)/2 Assuming that both X and G are connected, one can show that for n ≥ 2 this upper bound is attained only for X isometric to one of the following spaces of constant sectional curvature: the Euclidean space Rn, the sphere Sn, the hyperbolic space RHn and the projective space RPn, with the group G being isomorphic to the connected component of the isometry group of the corresponding model. 2 we review fundamental facts on proper actions starting with the topological setup and gradually progressing to actions of Lie groups on smooth manifolds by diffeomorphisms The latter are discussed, where a number of classification results for pairs (X, G) are presented beginning with the maximal group dimension n(n + 1)/2.

Fundamentals of proper actions

Proper actions of high-dimensional groups

The complex case

Kobayashi-hyperbolic manifolds