@Ted So I think the first part goes like this: let $\alpha$ be a holomorphic $(1,0)$-form, i.e. $\overline{\partial}\alpha=0$, then $\partial\alpha$ is a $(2,0)$-form and its conjugate $\overline{\partial}\overline{\alpha}$ is a $(0,2)$-form, so that $\partial\alpha\wedge\overline{\partial}\overline{\alpha}=\partial(\alpha\wedge\overline{\partial}\overline{\alpha})=d(\alpha\wedge\overline{\partial}\overline{\alpha})$ is a $(2,2)$-form whose integral vanishes by Stokes. On the other hand, locally we can write $\partial\alpha=fdz^1\wedge dz^2$ for some $f$ and $\overline{\partial}\overline{\a…
