CALABI-YAU 4-FOLDS AND TORIC FIBRATIONS

We present a general scheme for identifying fibrations in the framewor k of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d less than or equal to 150 whose maximal Newton polyhedra are reflexive and 535 577 weights w ith degree d less than or equal to 4000 that give rise to weighted pro jective spaces such that the polynomial defining a. hypersurface of tr ivial canonical class is transversal. We compute all Hodge numbers, us ing Batyrev's formulas (derived by toric methods) for the first and Va fa's fomulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k-folds, including the ellip tic cast, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is 3 subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.