we prove a version of the Lefschetz hyperplane theorem that allows one to find such Gysin morphisms even in some cases in which neither (a) nor (b) are satisfied. Key to our result are cubical hyperresolutions of varieties, which can be thought of as a way of resolving a variety ‘at all levels’. Indeed, when computing a resolution one usually stops when the exceptional divisor has simple normal crossings, but one could go on and resolve the singularities of the exceptional divisor and so on. Cubical hyperresolutions are a precise formalisation of this ‘inductive resolution’ procedure.

Cubical hyperresolutions were used in [@GNPP] to prove another version of the Lefschetz hyperplane theorem, which was the main motivation for our work. The precise statement is the following:

\[thm:Guillen\] [@GNPP Corollaire III.3.12] Let $X$ be a quasi-projective complex variety and $Y$ be a hyperplane section of $X$ satisfying the following hypotheses:

(i) There exists an augmented $m$-cubical hyperresolution $X_{{\square}}\rightarrow X$ such that $Y_{{\square}}:=X_{{\square}}\times_{X}Y$ is a cubical hyperresolution of $Y$.

(ii) For any ${\alpha}\in{\square}_m^*$, there exists a closed immersion $Y_{{\alpha}}\hookrightarrow X_{{\alpha}}$ of codimension $1$.

Then there exist Gysin morphisms between de Rham cohomology groups

$$H_{DR}^k(Y)\rightarrow H_{DR}^{k+2}(X)$$

which are isomorphisms for $k>\dim(Y)$ and a surjection 