Locally conformally Kähler manifolds admitting a holomorphic conformal flow

Abstract

A manifold $M$ is locally conformally Kähler (LCK) if it admits a Kähler covering $\tilde{M}$ with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on $\tilde{M}$. We show that $M$ admits an automorphic potential, and the monodromy group of its conformal weight bundle is $\mathbb{Z}$.