Automorphisms of Locally Conformally Kähler Manifolds

Abstract

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering ˜ M with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automor- phisms, an averaging procedure gives a G-invariant LCK metric. Suppose that S 1 acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover ˜ M is not isometric. We show that ˜ M admits an automorphic Kahler potential, and hence (for dimCM > 2) the manifold M can be embedded to a Hopf manifold.