holomorphic vector field on Fano Kähler-Einstein manifold

Question

Let M be a compact Fano Kähler-Einstein manifold, V is a holomorphic (1,0) vector field. Fano conditions tells that $V=\nabla^{1,0} f$ for some smooth complex valued function.By Matsushima theorem, Kähler-Einstein condition implies that $div V=\Delta f$ is an eigenfunction of complex laplacian $\Delta$. My question is would it be possible to take the potential fucntion $f$ to be a real valued function?

Answer 1

Actually, I just figured out an argument. Take any holomorphic $(1,0)$ vector field, say $X$, then Fano implies $X=\nabla^{1,0}f$, with complex valued $f$. And $\operatorname{div} X=\Delta f$. Now let $f=u+iv$, then since $\operatorname{div} X$ is a $1$-eigenfunction of the complex Laplacian, so are $\Delta u$ and $\Delta v$, then by Matsushima theorem, one knows that $\nabla^{1,0}(\Delta u)$ is also a holomorphic vector field Which has real potential $\delta u$. Since the Laplacian is an real operator in the Kähler case.