Effective Transitive Actions of the Unitary Group on Quotients of Hopf Manifolds

In our joint article with N. Kruzhilin of 2002, 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 $$\mathop {\text {U}}\nolimits _n$$ 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 $$\mathop {\text {U}}\nolimits _n$$ on such quotients, which provides an answer to a 10-year-old question.