Kahler manifolds with real holomorphic vector fields

For a Kähler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $Δ_{f}$ 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 Kähler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.

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