Proper group actions in complex geometry

TL;DR

This survey reviews the classification of proper Lie group actions on complex manifolds, emphasizing high-dimensional automorphism groups and Kobayashi-hyperbolic manifolds, highlighting recent advances in understanding their structure.

Contribution

It provides a comprehensive exposition of classification results for proper Lie group actions in complex geometry, including explicit descriptions of hyperbolic manifolds with large automorphism groups.

Findings

Complete classification of Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups

Explicit determination of all proper effective actions in high dimensions

Enhanced understanding of the structure of proper group actions in complex geometry

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.