Results from an algebraic classification Calabi-Yau manifolds

We present results from an inductive algebraic approach to the systematic c onstruction and classification of the 'lowest-level' CY3 spaces defined as zeroes of polynomial loci associated with reflective polyhedra, derived fro m suitable vectors in complex projective spaces. These CY3 spaces may be so rted into 'chains' obtained by combining lower-dimensional projective vecto rs classified previously. We analyze all the 4 242 (259, 6, 1) two- (three- , four-, five-) vector chains, which have, respectively, K3 (elliptic, line -segment, trivial) sections, yielding 174 767 (an additional 6 189, 1582, 1 99) distinct projective vectors that define reflective polyhedra and thereb y CY3 spaces, for a total of 182 737. These CY3 spaces span 10 827 (a total of 10 882) distinct pairs of Hodge numbers h(11), h(12). Among these, we l ist explicitly a total of 212 projective vectors defining three-generation CY3 spaces with K3 sections, whose characteristics we provide. (C) 2001 Pub lished by Elsevier Science B.V.