The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality

Abstract

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a d-dimensional Newton polytope ∆ are Calabi-Yau varieties X if and only if the Fine interior of the polytope ∆ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of such Calabi-Yau variety X via the lattice polytope ∆. This formula allows to test mirror symmetry in cases when ∆ is not a reflexive polytope. In particular, we apply this formula to pairs of lattice polytopes (∆, ∆ ∨ ) that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs (∆, ∆ ∨ ) show that the stringy Euler numbers of the corresponding Calabi-Yau varieties X and X ∨ may not satisfy the expected topological mirror symmetry test: e st (X) = (−1) d−1 e st (X ∨ ). This shows the necessity of an additional condition on Mavlyutov's pairs (∆, ∆ ∨ ).

All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra

“…(the first condition ∆ = [∆] just means that ∆ is a lattice polytope). Such polytopes are called almost reflexive [27] or pseudoreflexive [28] and give rise to well-defined singular varieties of Calabi-Yau type.…”

“…Structures with a band-like appearance as in Figs 15,18,27. and 30 are also related to (4) and(28): assuming that both h 1,2 and h 1,1 + h 1,3 have a preference for being even, χ/6 = 8 + h 1,1 − h 1,2 + h 1,3 and h 2,2 /2 = 22 + 2h 1,1 − h 1,2 + 2h 1,3…”

On the Fine Interior of Three-dimensional Canonical Fano Polytopes

“…Another interior of a lattice polytope ∆ was introduced by J. Fine [Fin83,Rei87,Ish99,Bat17]: Then the convex subset…”

“…In particular, ∆ FI is non-empty if ∆ • ∩ M is non-empty. Moreover, for any lattice polytope ∆ of dimension d ≤ 2 one has the equality conv(∆ • ∩ M ) = ∆ FI [Bat17]. The Fine interior ∆ FI of a lattice polytope ∆ of dimension d ≥ 3 may happen to be strictly larger than the convex hull conv(∆ • ∩ M ).…”

On smooth Calabi-Yau threefolds of Picard number two

“…For recent general results on boundedness of Calabi-Yau threefolds we refer to [13] as well as [38] for the case of Picard number two.Hypersurfaces in toric Fano varieties form a rich source of examples for Calabi-Yau varieties, e.g. [1,6,7]. Theorem 1.1 comprises several varieties of this type.Remark 1.2.…”