Multiplier Hermitian structures on Kähler manifolds

Abstract

The main purpose of this paper is to make a systematic study of a special type of conformally Kähler manifolds, called multiplier Hermitian manifolds, which we often encounter in the study of Hamiltonian holomorphic group actions on Kähler manifolds. In particular, we obtain a multiplier Hermitian analogue of Myers' Theorem on diameter bounds with an application (see [M5]) to he uniquness up to biholomorphisms of the "Kähler-Einstein metrics" in the sense of [M1] on a given Fano manifold with nonvanishing Futaki character.