Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds
Victor V. Batyrev, Lev A. Borisov
TL;DR
The paper develops a toric, combinatorial framework based on reflexive Gorenstein cones to realize mirror symmetry for generalized and rigid Calabi–Yau manifolds. By establishing a duality $\sigma oreve{\sigma}$ of cones (and the corresponding nef-partitions) it connects complete-intersection and hypersurface constructions, unifying known mirror constructions and extending them to generalized and rigid cases. It provides concrete mechanisms (via reductions to hypersurfaces, nef-partitions, and orbifold Gepner models) and evidence that the cone duality implements the mirror involution, with explicit calculations in the rigid CY example showing matching Hodge-theoretic data. The approach integrates toric Fano geometry, polyhedral dualities, and Jacobian-ring methods to yield a cohesive framework potentially applicable across Calabi–Yau, generalized CY, and rigid CY mirror pairs, offering new insights into the global structure of mirror symmetry.
Abstract
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.