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≥ 2 that admits an effective transitive action by holomorphic transformations of the unitary group U _n is biholomorphic to the quotient of a Hopf manifold by the action of 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 U _n on such quotients, which provides an answer to a 10-year-old question.