Holomorphic forms on a compact Kähler manifold are closed
= Holomorphic forms on a compact Kähler manifold are closed
A holomorphic $(p,0)$ form is annihilated by $\bar\partial$ and, by bidegree, by $\bar\partial^*$. The <Kähler Laplacian identity> makes it harmonic for $d$, hence $d$-closed on a compact <Kähler manifold>.