Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties
Lev Borisov
TL;DR
The paper addresses constructing mirror partners for Calabi–Yau complete intersections within Gorenstein toric Fano varieties by extending Batyrev's polar duality from hypersurfaces to complete intersections. It develops a combinatorial duality based on nef-partitions of reflexive polytopes, introducing related polytopes $\Delta_i$ and $\nabla_i$ and establishing the involutive relations $\Delta^* = \nabla_1 + \cdots + \nabla_r$ and $\nabla^* = \Delta_1 + \cdots + \Delta_r$. This duality induces dual nef-partitions and yields pairs of mirror-symmetric Calabi–Yau complete intersections in the toric varieties $P_{\Delta^*}$ and $P_{\nabla^*}$, conjecturally realizing the mirror involution in this broader setting. Overall, the work provides a purely combinatorial framework to construct and study mirror pairs beyond hypersurfaces, broadening the scope of mirror symmetry in toric geometry.
Abstract
We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.