Kähler manifolds with real holomorphic vector fields

Abstract

For a Kahler 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 Kahler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.