Families of Calabi–Yau hypersurfaces in Q-Fano toric varieties

Abstract

We provide a sufficient condition for a general hypersurface in a 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 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 ∆ 1 ⊆ ∆ 2 , where ∆ 1 and ∆ * 2 are canonical.{W = 0} ⊂ P(w)/G ←→ {W * = 0} ⊂ P(w * )/G * ,

Quasismooth hypersurfaces in toric varieties

“…LetX := Spec R(X) ∼ = C |Σ(1)| , J be the irrelevant ideal in R(X), i.e. the ideal generated by the monomials xσ := ρ ∈σ (1) x ρ , σ ∈ Σ, andX :=X − V (J). The Cl(X)-grading of R(X) induces an action of the quasitorus G = Spec C[Cl(X)] onX which preservesX.…”

“…Let σ ∈ Σ and γ ⊆ σ(1) as in the statement, k := |γ| and s := |M γ |. We can assume that ∆ ρ σ (L) is not empty for all ρ ∈ γ since the empty polytopes give zero columns in both A Mγ ,γ and A Mγ ,Σ (1) . We will now prove that the collection of polytopes {∆ ρ σ (L)} ρ∈γ is degenerate if and only if 2 rk(A Mγ ,γ ) > rk(A Mγ ,Σ (1) ).…”

“…In [1] Berglund-Hübsch-Krawitz duality has been generalized to give a duality between pairs of polytopes. We recall the main definitions and results.…”

“…Proof. By [1,Corollary 1.6] it is enough to prove that the origin is an interior point of P 1 . Assume the contrary, i.e.…”

“…Our motivation for the study of this regularity condition is the fact that quasismooth hypersurfaces with trivial canonical class give examples of Calabi-Yau varieties with canonical singularities [1]. In fact, quasismoothness appears as a regularity condition in the literature on mirror symmetry, for example it is required for the Berglund-Hübsch-Krawitz duality [3], [15].…”