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

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $\mathrm{\Delta }$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(\mathrm{\Delta },{\mathrm{\Delta }}^{\vee })$ that appear in the Mavlyutov’s generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov’s dual pairs $(\mathrm{\Delta },{\mathrm{\Delta }}^{\vee })$ 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: $e_{\mathrm{s}\mathrm{t}}(X)=(-1{)}^{d-1}{e}_{\mathrm{s}\mathrm{t}}(X^{\vee })$. This shows the necessity of an additional condition on Mavlyutov’s pairs $(\mathrm{\Delta },{\mathrm{\Delta }}^{\vee })$.