A proof of the birationality of certain BHK-mirrors

Clarke, Patrick

doi:10.2478/coma-2014-0003

Cited by 14 publications

(8 citation statements)

References 9 publications

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“…For the Batyrev-Borisov double mirrors, the equality of their stringy Hodge numbers is a consequence of the main result of [BB96] (Theorem 4.15), their derived equivalence are confirmed for the corresponding stacks in [FK16b] (Theorem 6.3), and their birationality has been proved under some mild assumptions in [Li13] (Theorem 4.10). The analogous results for Berglund-Hübsch-Krawitz mirrors have been established in [CR11,FK16a,Sho14,Bor13,Kel13,Cla14].…”

Section: Introductionsupporting

confidence: 70%

On derived equivalence of general Clifford double mirrors

2016

Preprint

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Section: Introductionsupporting

confidence: 70%

On derived equivalence of general Clifford double mirrors

2016

Preprint

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“…For instance, it may happen that two different almost pseudoreflexive polytopes ∆ 1 = ∆ 2 have the same pseudoreflexive duals, i.e., [∆ * 1 ] = [∆ * 2 ]. This equality is the key observation for the birationality of BHK-mirrors investigated in[Ke13,Cla14,Sh14].…”

mentioning

confidence: 70%

“…. This equality is the key observation for the birationality of BHK-mirrors investigated in [Ke13,Cla14,Sh14]. Definition 3.10.…”

Section: The Mavlyutov Dualitymentioning

confidence: 99%

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

Batyrev

2017

Pure and Applied Mathematics Quarterly

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

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“…The equivalence between orbifold Chen-Ruan cohomology of corresponding DM-stacks [Z A,G ] and [Z A ′ ,G ] is a consequence of the main result of [CR11]. The birationality of double mirrors is established in various generality in [Sho12,Kel13,Cla13,Bor13]. In [FK14], the derived categories of double mirrors are shown to be equivalent.…”

Section: Mirrors and Double Mirrorsmentioning

confidence: 99%