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Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology

$$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$

Can we say anything similar if $X$ is non-compact, for example if $X=R\setminus S$ for some compact Riemann surface $R$ and a (finite) set of points $S$?

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    $\begingroup$ Noncompact Riemann surfaces are Stein manifolds, so in positive degrees their cohomology with respect to any coherent analytic sheaf vanishes. The left hand side of your equation is thus always zero, but the right hand side is often nonzero. $\endgroup$
    – Andy Putman
    Commented Mar 26 at 13:30
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    $\begingroup$ I thought I'd mention that Serre's original result in "Un théorème de dualité" Commentarii 1955, applies equally well to noncompact manifolds. The other side is the dual of compactly supported cohomology viewed as a Frechet space. The interesting statement in this case would be that $H^0(X,E)=H_c^1(X,\Omega\otimes E^*)^*$. So the question doesn't seem so bad to me. $\endgroup$
    – Donu Arapura
    Commented Mar 26 at 18:45
  • $\begingroup$ @DonuArapura: I agree, and I did not intend my comment to be construed as saying it was a bad question (the one upvote is mine too!). I just wanted to make some trivial remarks in case the OP didn’t know them, and figured someone that was more of an expert would eventually give a real answer. $\endgroup$
    – Andy Putman
    Commented Mar 27 at 2:32

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