Let $X$ be a toric Calabi-Yau manifold, of complex dimension $d=4$. What do we know about its cohomology? In particular, if we know $H^{1,1}(X)$ (say we know its dimension and pick a basis), can we say something about $H^{2,2}(X)$ and $H^{3,3}(X)$?
Clarification: let $\mathbb C^{N+4}$ with coordinates $z_a$, and $U(1)^N$ action $z_a \to e^{\sqrt{-1} \alpha_i Q^i_a} z_a$ for some matrix of integers $Q$ satisfying $\sum_{a=1}^{N+4} Q^i_a = 0$ for all $i=1,\ldots,N$, and moment maps $$ \mu_i = \sum_a Q^i_a |z_a|^2 - r_i : \mathbb C^{N+4} \to \mathbb R$$ for real numbers $r_i$. Then $X= \mu^{-1}(r) / U(1)^N$.