= 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.
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