On a class of complete intersection Calabi–Yau manifolds in toric manifolds

We consider the family of smooth $n$-dimensional toric manifolds generalizing the family of Hirzebruch surfaces to dimension $n$. We analyze conditions under which there exists a Calabi–Yau complete intersection of two ample hypersurfaces in these manifolds. This turns out to be possible only if the toric manifold is the product of projective spaces. If one of the hypersurfaces is not ample then we find Calabi–Yau complete intersection of two hypersurfaces in Fano manifolds of the given family.