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Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{D})=0?$$

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  • $\begingroup$ You mean $H^1(\mathbb D, \mathcal O_{\mathbb D}(D))$, right? Or what do you mean? $\endgroup$
    – Will Sawin
    Commented Apr 8, 2016 at 3:54
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    $\begingroup$ Assuming you mean $\mathcal{O}_{\mathbb{D}}(D)$: any divisor on $\mathbb{D}$ is principal, so $\mathcal{O}_{\mathbb{D}}(D)$ is isomorphic to $\mathcal{O}_{\mathbb{D}}$, and of course its higher cohomology is trivial. $\endgroup$
    – abx
    Commented Apr 8, 2016 at 4:26

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