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

TL;DR

This paper provides a combinatorial formula for the stringy Euler number of Calabi-Yau hypersurfaces in toric varieties, explores Mavlyutov duality, and examines mirror symmetry beyond reflexive polytopes.

Contribution

It introduces a new formula for stringy Euler numbers applicable to non-reflexive polytopes and analyzes the limitations of mirror symmetry in Mavlyutov dual pairs.

Findings

The formula enables testing mirror symmetry in broader cases.

Examples show stringy Euler numbers may not satisfy expected mirror symmetry.

Additional conditions are needed for Mavlyutov pairs to satisfy mirror symmetry.

Abstract

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