As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on 'analytic' methods (in particular, on stratified Morse theory). This probably means that there exist certain facts whose proof is known in characteristic 0 only. Is this true (at the moment)?
In particular, I would like to know, whether the following cohomological analogue of Lazarsfeld's Theorem 3.5.11 is known (in positive characteristic).
Let $X$ be a projective local complete intersection variety over an algebraically closed field of characteristic $p$ (that is positive); $f:X\to P^r$ is a finite morphism. Suppose that $Y\subset P^r$ is a closed local complete intersection subvariety of pure codimension $d$. Then the induced homomorphism $f^*:H^i(P^r,Y)\to H^i(X,f^{-1}(Y))$ is an isomorphism when $i\le n-d$.
Here $H^\ast$ denotes the (relative) etale cohomology with $\mathbb{Z}/l\mathbb{Z}$-coefficients, where $l$ is prime to p. I hope that I passed from homotopy (as in Lazarsfeld's statement) to cohomology (in 'my' version above) correctly. Any comments would be very welcome!
