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In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems like he is using that $$\int_{\partial X}\theta=0,$$ and since $X$ might have boundary, how do we know that this is true? Is it even true? He makes a ton of small mistakes, so he might have meant to say a closed Riemann surface.

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