Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds

We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with central chatge $c = 3d$. It is conjectured that the natural combinatorial duality satisfies by these cones corresponds to the mirror involution. Using the theory of toric varieties, we show that our conjecture includes as special cases all already known examples of mirror pairs proposed by physicists and agrees with previous conjectures of the authors concerning explicit constructions of mirror manifolds. In particular we obtain a mathematical framework which explains the construction of mirrors of rigid Calabi-Yau manifolds.