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?

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    $\begingroup$ Crossposted on MSE. $\endgroup$ – Michael Albanese Jun 8 '18 at 19:52
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    $\begingroup$ I don't understand the problem. For any compact complex manifold, the space of holomorphic vector fields is finite-dimensional -- for instance because it is the space of global sections of a holomorphic vector bundle. $\endgroup$ – abx Jun 8 '18 at 19:53
  • $\begingroup$ You cannot check co-closedness by a local calculation, because it's not true locally. You have to use compactness of the manifold, and specifically the Hodge decomposition theorem (see e.g. Theorem 4.1.13 in Huybrechts' Complex Geometry), which immediately gives what you want. This is just an elaboration on what abx said, of course. $\endgroup$ – YangMills Jun 9 '18 at 8:58

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