isomorphisms for $k>\dim(Y)$ and a surjection for $k=\dim(Y)$.

2.  if $c=0$ then the conclusion of point 1. holds for $k>2\dim(\Sigma_X)+1$.

For the terminology regarding cubical hyperresolutions and cubical varieties, as well as for the definition of (algebraic) de Rham cohomology groups, we refer to Section 1.

Observe that if $c=1$ then we obtain the same Gysin morphisms provided by the usual Lefschetz hyperplane theorem; one can check that if (b) holds then we are in this situation.

We were able to find an example in which the hypotheses of Theorem \[thm:risultato\] are satisfied, but (a), (b) and those of Theorem \[thm:Guillen\] are not. Our example relies on the fact that both $X$ and $Y$ have isolated and very simple singularities, which makes it easy to control the cubical hyperresolutions; this suggests that as the dimension and complexity of $\Sigma_X$ and $\Sigma_Y$ increase finding examples in which conditions (i) and (ii) of Theorem \[thm:risultato\] are satisfied becomes much harder.

Using Theorem \[thm:risultato\] we were able to prove the triviality of the Alexander polynomial of a particular class of line arrangements in ${\mathbb{P}}^2$. The study of the Alexander polynomial of line arrangements is a vast subject, lying at the crossroads of topology, 