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Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily smooth!!) subvarieties of $X$ of dimension $m$ (here $H^\ast$ is singular or etale cohomology)? Did anybody study this filtration (for fixed $X,i$, when $m$ varies) before? What properties can one prove for it?

Some remarks.

  1. If $Z$ is a generic multiple hyperplane section of $X$, $i\le m$, then a theorem of Beilinson yields that the map $H^i(X)\to H^i(Z)$ is injective.

  2. On the other hand, I believe that $H^{m+1}(X)\to H^{m+1}(Z)$ cannot be injective if $X$ is 'too complicated'; yet I have no idea how to construct any more or less general examples here.

  3. Certainly, it suffices to consider only 'large enough' $Z$ here.

  4. The filtration in question is closely related with the one given by $\cap Ker (H^i(f):H^i(X)\to H^i(Z))$ for $f$ running through proper morphisms with $Z$ of dimension $m$ (and smooth).

  5. If $X$ is proper, then one can replace $H^\ast$ here with $H_c^\ast$.

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