Effective transitive actions of the unitary group on quotients of Hopf manifolds

TL;DR

This paper classifies all effective transitive actions of the unitary group on quotients of Hopf manifolds, providing explicit descriptions and answering a longstanding open question in complex geometry.

Contribution

It offers a complete explicit classification of effective transitive ${ m U}_n$ actions on Hopf manifold quotients, extending previous results from 2002.

Findings

Explicit description of all effective transitive ${ m U}_n$ actions on Hopf manifold quotients

Resolution of a 10-year-old open problem in complex geometry

Extension of previous classification results from 2002

Abstract

In our article of 2002 joint with N. Kruzhilin we showed that every connected complex manifold of dimension $n\ge 2$ that admits an effective transitive action by holomorphic transformations of the unitary group ${\rm U}_n$ is biholomorphic to the quotient of a Hopf manifold by the action of ${\mathbb Z}_m$ for some integer $m$ satisfying $(n,m)=1$. In this note, we complement the above result with an explicit description of all effective transitive actions of ${\rm U}_n$ on such quotients, which provides an answer to a 10-year old question.