Families of Calabi-Yau hypersurfaces in $\mathbb Q$-Fano toric varieties

TL;DR

This paper establishes criteria for Calabi-Yau hypersurfaces in $Q$-Fano toric varieties, introduces a generalized mirror construction, and explores a duality framework unifying Batyrev and Berglund-H"ubsch-Krawitz methods.

Contribution

It provides a sufficient condition for Calabi-Yau hypersurfaces, generalizes the Berglund-H"ubsch-Krawitz construction, and introduces a duality linking different mirror symmetry approaches.

Findings

Criteria for Calabi-Yau hypersurfaces in $Q$-Fano toric varieties

A generalized mirror construction unifying existing methods

A duality framework based on polar duality of polytopes

Abstract

We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-H\"ubsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes $\Delta_1\subseteq \Delta_2$, where $\Delta_1$ and $\Delta_2^*$ are canonical.