The situation is this:
I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have been able to compute the dimensions of the sheaf cohomology groups $H^i(Y,\Omega_{Y/k}^1)$ for $i=1,2$, with the help of Macaulay2 (these are $1$ and $13$-dimensional respectively), and a few other invariants, such as the normal bundle $h^i(Y,\mathcal N_{Y/\mathbb P^{13}})$.
Let $X$ be the smooth variety obtained by intersecting $Y$ with two general hyperplanes $H_1,H_2$ in $\mathbb P^{13}$. Then $X$ is $3$-dimensional, and I can also prove that it is Calabi-Yau. Now computations are not possible by computer anymore, because I no longer have any nice Gröbner basis.
The question is this: What can I deduce about the Hodge numbers $h^{1q}=h^q(X,\Omega_{X/\mathbb C}^1)$ of $X$ given my knowledge about $H^i(Y, \Omega^1_{Y/\mathbb C})$ (and possibly other properties of $Y$?)? Is there some Lefschetz type theorems with weaker hypotheses (the usual hypothesis of Lefschetz is smoothness of $Y$), for example?