A theorem of Calabi-Matsushima's type (Q1598259)

An analogue of the theorems of \textit{E. Calabi} [in Differential geometry and complex analysis, 95-114 (1985; Zbl 0574.58006)] and \textit{Y. Matsushima} [Holomorphic vector fields on compact Kähler manifolds, Regional Conf. Ser. Math. 7 (1971; Zbl 0218.53084)] is here proved for a special type of conformally Kähler manifolds, satisfying a suitable Einstein condition, the so-called multiplier Hermitian manifolds. The non-singularity of multipliers plays a central role in the given proof.