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

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Abstract: We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a

d

-dimensional Newton polytope

D e l t a

are Calabi-Yau varieties

X

if and only if the Fine interior of

D e l t a

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

D e l t a

is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes

(D e l t a, D e l t a^{v e e})

that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs

(D e l t a, D e l t a^{v e e})

show that the stringy Euler numbers of the corresponding Calabi-Yau varieties

X

and

X^{v e e}

may not satisfy the expected topological mirror symmetry test:

e_{m s t} (X) = (- 1)^{d - 1} e_{m s t} (X^{v e e})

. This shows the necessity of an additional condition on Mavlyutov's pairs

(D e l t a, D e l t a^{v} e e)

.

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