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Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets of $D_i$ are smooth; take $H$ either be the etale cohomology theory (with certain coefficients) or singular cohomology (if $k$ is the field of complex numbers).

My question is: in which cases of this setting one has considered the Leray spectral sequence corresponding to the morphism $U\to X$ and converging to $H^*(U)$? In which cases it was proved that this spectral sequence is the Deligne's weight one?

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  • $\begingroup$ I suspect you may need to assume further that the $D_i$ meet transversally to get a good result of this type. Let me give it some thought. $\endgroup$
    – Donu Arapura
    Commented Sep 9, 2021 at 7:09
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    $\begingroup$ In my paper "A spectral sequence for stratified spaces and configuration spaces of points" I construct a spectral sequence associated to this situation, which in many situations (but not always) coincides with the Leray spectral sequence. I'm quite sure that if the poset of all intersections of the $D_i$ is a Cohen-Macaulay poset, graded by codimension, then my spectral sequence coincides with the Leray spectral sequence and gives Deligne's weight filtration on cohomology. $\endgroup$
    – Dan Petersen
    Commented Sep 9, 2021 at 7:34
  • $\begingroup$ Dan, perhaps you can turn your comment into an answer. $\endgroup$
    – Donu Arapura
    Commented Sep 9, 2021 at 12:14

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