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

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.

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