I am trying to compute the Hodge diamond of a Calabi-Yau fourfold which is a complete intersection inside of a projective bundle over a threefold base. I have computed the arithmetic genera $\chi\_1$ and $\chi\_2$ which give me two linear relations between the four independent Hodge numbers $h^{(1,1)}$, $h^{(1,2)}$, $h^{(1,3)}$ and $h^{(2,2)}$. So if I can compute two of them the arithmetic genera will give me the other two. Since my variety is a Calabi-Yau fourfold, $h^{(1,1)}$ is the second Betti number and $h^{(1,2)}$ is twice the third Betti number, so really all I need are the second and third Betti numbers to finish the calculation. As such, I wanted to use the Lefschetz hyperplane theorem to relate the cohomology of my fourfold to the cohomology of the ambient projective bundle, but it turns out that the divisors for which my fourfold is a complete intersection of are not ample (they are only relatively ample). So does anyone have any ideas on how to relate the cohomology of my variety to the cohomology of the projective bundle if the divisors for which my variety is a complete intersection of are only relatively ample? 



Hodge diamond of a Calabi-Yau fourfold
I encountered a similar problem a few years ago, but then in dimension 3. In that case a master student wanted to calculate the hodge diamond of a threefold which was a hypersurface $W$ in a $P^2$-bundle over a del Pezzo surface. This threefold was birational to a singular hypersurface $Y$ in some weighted projective 4-space. 
Now you can apply Lefschetz' hyperplane theorem for $Y$, but it only works for the lower cohomology groups, since Poincar\'e duality might fail. There are methods to calculate the higher cohomology groups if the singularities of $Y$ are nice enough.


Then you consider a factorization of the birational map $W\dashrightarrow Y$, calculate exceptional divisors etc, in order to figure out the difference between the Hodge numbers of $W$ and of $Y$.


In the end this turned out to be too hard for the student in question, so I worked out some of the details. This particular example is worked out at the end of the first version (on arXiv) of my paper with Klaus Hulek on MW-groups of elliptic threefolds. The referee considered this example superfluous, therefore we omitted this example in the later versions and the published version of this paper.


