Towards an algebraic classification of Calabi-Yau manifolds I: Study of K3spaces

We present an inductive algebraic approach to the systematic construction a nd classification of generalized Calabi-Yau (CY) manifolds in different num bers of complex dimensions based on Batyrev's formulation of CY manifolds a s toric varieties in weighted complex projective spaces associated with ref lexive polyhedra. We show how the allowed weight vectors in lower dimension s may be extended to higher dimensions, emphasizing the roles of projection and intersection in their dual description and the natural appearance of C artan-Lie algebra structures. The 50 allowed extended four-dimensional vect ors may he combined in pairs (triples) to form 22 (4) chains containing 90 (91) K3 spaces, of which 94 are distinct, and one further K3 space is found using duality. In the case of CY3 spaces, pairs (triples) of the 10270 all owed extended vectors yield 4242 (259) chains with K3 (elliptic) fibers con taining 730 additional K3 polyhedra. A more complete study of CY3 spaces is left for later work.