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If you have a smooth projective variety over a field and a Weil cohomology theory, weak Lefschetz gives you some cohomological information about its smooth hyperplane sections. Now, if I give you a smooth proper variety, can you give me any cohomological information about its smooth closed subvarieties of codimension 1?

The question is admittedly somewhat ambiguous. I just know that if you smart enough, you can sometimes bootstrap from "good" cases. For example, Deligne has proved that odd Betti numbers of smooth proper varieties over $\mathbb{C}$ are even this way. Another example is when you have a proper, non-smooth variety, you can find a smooth proper hypercovering and get mixed Hodge structure.

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    $\begingroup$ Even if you start with a projective variety, I don't know if you can get all that much information about cohomology of arbitrary smooth closed subvarieties of codimension $1$. For instance, let $X$ be the blow-up of any smooth projective variety at a point, and let $E$ be the exceptional divisor. The cohomology of $X$ may be very complicated, but $E$ sees almost none of that complexity. $\endgroup$ – dhy Apr 20 at 2:20
  • $\begingroup$ One can probably say a lot more about the cohomology of a smooth codimension 1 subvariety $D \subset X$ given information about the cohomology of $X$ and $X \setminus D$. One way to think about the Lefschetz hyperplane section theorem is that there's a Gysin sequence involving the cohomology of $D$, $X$ and $X \setminus D$ and when $D$ is ample there are vanishing results for the cohomology of $X \setminus D$. $\endgroup$ – cgodfrey Apr 21 at 6:21

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