 Holomorphic p-form on a complex manifold

A differential form of type (p, q) of type $(p,0)$ is holomorphic when its local coefficients in holomorphic coordinates are holomorphic functions, equivalently $\bar{\partial }\alpha =0$. It need not be $d$-closed on every complex manifold. Top-degree holomorphic forms are closed by bidegree. On a compact complex surface, holomorphic forms of all degrees are closed, even without a Kähler metric.