Holomorphic forms on a compact complex surface are closed

On a compact complex surface, the claim also holds without a Kähler metric. For a holomorphic one-form $\alpha$, let $\beta =d\alpha$. It is a closed holomorphic two-form, and the Stokes theorem gives $\int \beta \wedge \bar{\beta }=\int d(\alpha \wedge \bar{\beta })=0$. Positivity of $\beta \wedge \bar{\beta }$ forces $\beta =0$. Functions are constant on each connected component, and holomorphic two-forms are closed by dimension.