Automorphisms of locally conformally Kahler manifolds

TL;DR

This paper investigates the symmetry properties of locally conformally Kahler (LCK) manifolds, showing that certain group actions imply the existence of automorphic Kahler potentials and embeddings into Hopf manifolds.

Contribution

It demonstrates that U(1) actions with non-isometric lifts on LCK manifolds lead to automorphic Kahler potentials and embeddings into Hopf manifolds, advancing understanding of LCK symmetry structures.

Findings

U(1) actions induce automorphic Kahler potentials

LCK manifolds with certain symmetries embed into Hopf manifolds

Averaging yields G-invariant LCK metrics

Abstract

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler 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 U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.