Kähler manifolds with real holomorphic vector fields

For a Kähler manifold endowed with a weighted measure (Formula presented.), the associated weighted Hodge Laplacian (Formula presented.) maps the space of (Formula presented.)-forms to itself if and only if the (Formula presented.)-part of the gradient vector field (Formula presented.) is holomorphic. We use this fact to prove that for such (Formula presented.), a finite energy (Formula presented.)-harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for(Formula presented.)-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such (Formula presented.)-harmonic maps must be constant if (Formula presented.) has an isolated minimum point. In particular, this implies that for a compact Kähler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.