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

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-Hübsch-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-Hübsch-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 ${\Delta }_2^{\ast }$ are canonical.

Final version, to appear in J. Math. Pures Appl

References in corpus (1)

• Canonical toric Fano threefolds