Mirror symmetry for Calabi-Yau hypersurfaces in weighted P₄ and extensions of Landau-Ginzburg theory

Recently two groups have listed all sets of weights k = (k_l,..., k₅) such that the weighted projective space P₄^k admits a transverse Calabi-Yau hypersurface. It was noticed that the corresponding Calabi-Yau manifolds do not form a mirror symmetric set since some 850 of the 7555 manifolds have Hodge numbers (b₁₁, b₂₁) whose mirrors do not occur in the list. By means of Batyrev's construction we have checked that each of the 7555 manifolds does indeed have a mirror. The 'missing mirrors' are constructed as hypersurfaces in toric varieties. We show that many of these manifolds may be interpreted as non-transverse hypersurfaces in weighted P₄'s, i.e. hypersurfaces for which dp vanishes at a point other than the origin. This falls outside the usual range of Landau-Ginzburg theory. Nevertheless Batyrev's procedure provides a way of making sense of these theories.