Results from an algebraic classification Calabi–Yau manifolds

We present results from an inductive algebraic approach to the systematic construction and classification of the ‘lowest-level’ CY3 spaces defined as zeroes of polynomial loci associated with reflexive polyhedra, derived from suitable vectors in complex projective spaces. These CY3 spaces may be sorted into ‘chains’ obtained by combining lower-dimensional projective vectors 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, 1 582, 199) distinct projective vectors that define reflexive polyhedra and thereby CY3 spaces, for a total of 182 737. These CY3 spaces span 10 827 (a total of 10 882) distinct pairs of Hodge numbers h11,h12. Among these, we list explicitly a total of 212 projective vectors defining three-generation CY3 spaces with K3 sections, whose characteristics we provide.