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



Holomorphic vector field on Fano Kähler–Einstein manifold
actually, I just figure out an argument. Take any hol'c (1.0) vector field, say X, then Fano implies $X=\nabla^{1,0}f$, with complex valued f. And $div X=\Delta f$. Now let f=u+iv, then since div X is 1-eigenfunction of complex Laplacian, so is \Delta u and \Delta v, then by matsushima theorem, one knows that $\nabla^{1,0}(\Delta u)$ is also a hol'c v.f. Which has real potential $\delta u$. since the laplacian is an real operator in the kahler case.


