= Solution
A <holomorphic differential form> $\alpha$ of type $(p,0)$ satisfies $\bar\partial\alpha=0$. The adjoint $\bar\partial^*$ lowers antiholomorphic degree, so $\bar\partial^*\alpha=0$ as well. Therefore $\Delta_{\bar\partial}\alpha=0$.
The <Kähler Laplacian identity> gives $\Delta_d\alpha=2\Delta_{\bar\partial}\alpha=0$. <Compactness> lets us use the $L^2$ identity for the <Hodge Laplacian>:
$$
0=\langle\Delta_d\alpha,\alpha\rangle
=\|d\alpha\|^2+\|d^*\alpha\|^2.
$$
Both terms are nonnegative, so
$$
\boxed{d\alpha=0.}
$$
This proves that <holomorphic forms on a compact Kähler manifold are closed>. The original PDF uses $\Omega_M^p$ and $\mathcal A_{M,\mathbb C}^{p+1}$; the extra symbols in the local TeX superscripts are transcription errors.
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