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

Understand

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.