I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector fields $\mathfrak{h}$ is finite-dimensional, he argues the following.

Firstly $\mathfrak{h}$ is identified with the space of holomorphic vector fields of $T^{1,0}$, which is equipped with an obvious $\overline{\partial}$ operator. Then he claims that holomorphic vector fields belong to $Ker(\overline{\partial})$ and $Ker(\overline{\partial}^*)$, being $\overline{\partial}^*$ the adjoint of $\overline{\partial}$, with respect to any compatible metric. Finally, since the kernel of the elliptic operator $\overline{\partial}^*$ + $\overline{\partial}$ is clearly finite dimensional, the result follows.

I am not able to prove that an holomorphic vector field $X$ belongs to $Ker(\overline{\partial}^*)$. When I identify $X$ with a (0,1)-form $\omega$, using the Kähler metric, it is not hard to prove that $\omega$ is $\overline{\partial}$-closed. However, expressing $\omega$ in local coordinates, I am not able to derive also the $\overline{\partial}^*$-closedness.

Do you have any suggestion?