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Assume the following is given:

  • $X$ quasi-projective (smooth) variety,
  • $\overline X$ a projective variety, such that $X$ is the complement of a divisor $D$ in $X$, and
  • $\mathcal F$ a coherent sheaf on $\hat X$.

Suppose, we know the cohomology $H^\ast(X,{\mathcal F}|_X)$ of $\mathcal F$ on $X$. Now, for sure, this is not enough to be able to compute $H^{\ast}({\overline X},{\mathcal F})$. My question is what kind of additional data (localized near D) do we need to know in order to be able to compute $H^\ast({\overline X},{\mathcal F})$?

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    $\begingroup$ This is the subject of local cohomology: see e.g. SGA II. The answer (not far from tautologically) is that you need to know (derived) global sections with support in the complement $D$. $\endgroup$
    – Moosbrugger
    Commented Jul 18, 2011 at 15:49

1 Answer 1

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You need the cohomology of the extension of $\mathcal{F}$ to the completion of $X$ along $D$ (a formal scheme). There is an equivalence of categories that you can look up in

Moret-Bailly, Laurent Un problème de descente. Bulletin de la Société Mathématique de France, 124 no. 4 (1996), p. 559-585 (numdam)

see also references therein.

From this description, a way of computing cohomology should follow.

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