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 $\Delta$ are Calabi–Yau varieties $X$ if and only if the Fine interior of the polytope $\Delta$ 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 $\Delta$. This formula allows us 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: $e_{\mathrm{st}} (X) = (-1)^{d-1} e_{\mathrm{st}} (X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov’s pairs $(\Delta ,\Delta ^{\vee})$.

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Received 13 July 2017

Published 14 September 2018