Multiplier Hermitian structures on Kahler manifolds

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