Can one formulate those version of Weak Lefschetz that uses tubular neighbourhoods purely in terms of cohomology of (some) algebraic varieties? Theorem in 5.1 of Part II in Goresky-MacPherson's "Stratified Morse theory" implies (in particular) that: for a smooth projective P (over the field of complex numbers), X open in P, and a small enough tubular neighbourhood $H_\delta$ of an arbitrary (!) hyperplane section H of P (in P!) a Weak Lefschetz theorem for $(H_\delta\cap X,X)$ is valid i.e.: the map on singular cohomology $H^{i}_{sing}(X)\to H^{i}_{sing}(H_\delta\cap X)$ is an isomorphism for $i<\dim X-1$, and is an injection for $i=\dim X-1$. A caution: $H_\delta\cap X$ is not (usually) a tubular neighbourhood of $H\cap X$ in $X$.

My question is: could one formulate an analogue of this statement purely in terms of algebraic geometry? I would be completely satisfied with cohomology with $Z/l^n Z$-coefficients i.e. etale cohomology. I only want to replace the cohomology of $H^{i}_{sing}(H_\delta\cap X)$ in the statement by something that could be computed without using differential geometry.

My guess: one should probably replace $H_\delta$ with an etale tubular neighbourhood of $H$ in $P$ (then $H_\delta\cap X$ will be replaced by the corresponding fibre product); this is 'my conjecture'. Etale tubular neighbourhoods were defined and studied by Cox and Friedlander. Yet though they proved that etale tubular neighbourhoods share several properties with 'ordinary' tubular neighbourhoods, it seems that no comparison statement that would allow to deduce my conjecture from the Goresky-MacPherson's theorem is known. One should probably use nice properties of the comparison of the etale site with the fine one; yet this seems to require a site-theoretic definition of a tubular neighbourhood. Also, etale tubular neighbourhoods seem to be rather 'implicit', so I don't know how to check my conjecture on examples. Certainly, I do not object against proving my conjecture 'directly', yet this seems to be difficult (since Goresky-MacPherson's proof heavily relies upon stratified Morse theory).

Any suggestions would be very welcome!

  • $\begingroup$ You might want to look into the embedding into the normal cone. I know that it acts a lot like a tubular neighborhood in some ways, and it's rather helpful in many intersection theoretic arguments. Fulton describes it in his book "Intersection Theory" $\endgroup$ Nov 30, 2009 at 13:05
  • $\begingroup$ Unfortunately, the deformation of the normal cone construction does not seem to help. Roughly, my problem is to find a 'nice' way to 'lift' 'large enough' closed subvarieties of H (this corresponds to $H\setminus (H\cap X)$ in my question) of codimension c to subvarieties of P of the same codimension. In differential geometry one can do this (for submanifolds) using tubular neighbourhoods. Andd it seems that the deformation of the normal cone increases the codimension. Yet thanks; maybe I should think about this further! $\endgroup$ Nov 30, 2009 at 13:56

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regarding the comparison theorem between (Z/l^n-cohomology of) etale and complex-analytic tubular neighborhoods, my feeling is that it should hold, but won't follow formally from the comparison theorem for varieties.

one can at least get the comparison map, i think, as follows (this also addresses your question about sites): both in the etale and complex-analytic cases, the intersection of an open subset U of X with a tubular neighborhood of a closed subvariety Z in X can be represented by the topos which is the lax 2-fiber product of Sh(Z) and Sh(U) over Sh(X). so the comparison map of topoi comes from ordinary considerations. but to show that it's an isomorphism on cohomology, one probably should argue as in the proof of artin's comparison theorem, locally fibering by curves, etc. hopefully this vague sketch works out, but i definitely haven't considered the details.

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    $\begingroup$ Yes, I have thought about some 'intermediate' site here; this could help (thank you for supporting this idea:)). Do you know of any paper where such a site was considered? $\endgroup$ Nov 30, 2009 at 22:13

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