Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 5 / a / Solution

A holomorphic differential form $\alpha$ of type $(p,0)$ satisfies $\bar{\partial }\alpha =0$. The adjoint ${\bar{\partial }}^{\ast }$ lowers antiholomorphic degree, so ${\bar{\partial }}^{\ast }\alpha =0$ 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:Both terms are nonnegative, soThis 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.