Harmonic-Killing vector fields on Kähler manifolds

In a previous paper we have considered the harmonicity of local infinitesimal transformations associated to a vector field on a (pseudo)-Riemannian manifold to characterise intrinsically a class of vector fields that we have called harmonic-Killing vector fields. In this paper we extend this study to other properties, such as the pluriharmonicity and the α-pluriharmonicity (α harmonic 2-form) of the local infinitesimal transformations, obtaining characterisations of these kinds of vector fields.