Mirror Symmetry for Calabi – Yau Hypersurfaces in Weighted IP 4 and Extensions of Landau – Ginzburg Theory *

Recently two groups have listed all sets of weights k = (k1, . . . , k5) such that the weighted projective space IP4 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 IP4’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. * Supported in part by the Robert A. Welch Foundation, the Swiss National Science Foundation, a WorldLab Fellowship, N.S.F. grants DMS-9311386 and PHY-9009850 and a grant from the Friends of The Institute for Advanced Study.