Kahler manifolds with real holomorphic vector fields

TL;DR

This paper investigates conditions under which weighted Kähler manifolds with real holomorphic vector fields exhibit specific harmonic and pluriharmonic properties, revealing implications for harmonic maps and fundamental group homomorphisms.

Contribution

It establishes that the weighted Hodge Laplacian preserves forms under certain holomorphic gradient conditions and explores the implications for harmonic functions and maps.

Findings

Weighted Hodge Laplacian preserves forms if the (1,0)-part of ∇f is holomorphic.

Finite energy f-harmonic functions are pluriharmonic under these conditions.

Such f-harmonic maps into negatively curved manifolds are constant if f has an isolated minimum.

Abstract

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.