Automorphisms of locally conformally Kahler manifolds
A manifold M is locally conformally Kähler (LCK), if it admits a Kähler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that S1 acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kähler cover is not isometric. We show that admits an automorphic Kähler potential, and hence (for dimℂ M > 2) the manifold M can be embedded to a Hopf manifold.