  For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the
associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of
$(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector
field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a
finite energy $f$ harmonic function must be pluriharmonic. Motivated by this
result, we verify that the same also holds true for $f$-harmonic maps into a
strongly negatively curved manifold. Furthermore, we demonstrate that such
$f$-harmonic maps must be constant if $f$ has an isolated minimum point. In
particular, this implies that for a compact K\"{a}hler manifold admitting such
a function, there is no nontrivial homomorphism from its first fundamental
group into that of a strongly negatively curved manifold.
