2014
DOI: 10.1007/s00229-014-0693-7
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A note on bimodal singularities and mirror symmetry

Abstract: We discuss the relation between transposition mirror symmetry of Berlund and Hübsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter-Dynkin diagrams of bimodal singularities by Ebeling and Ploog.

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Cited by 10 publications
(36 citation statements)
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References 17 publications
(15 reference statements)
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“…Since the polar dual polytopes ∆ * 1 and ∆ * 2 of ∆1 and ∆2, and∆ * 3 and ∆ * 3 of∆3 and ∆3 are respectively the convex hulls of vertices (−1, −1, −1), (1, 0, 0), (1,3,9), (1, 3, −1), and (1, 0, −1), resp. (−1, −1, −1), (1, 1, 3), (1,3,9), (1, 3, −1), and (1, 0, −1), resp. (−1, −1, −1), (1, 0, −1), (1, 3, 11), (1, 3, 1), and (1, 0, −2), resp.…”
Section: No 38-no 40mentioning
confidence: 99%
“…Since the polar dual polytopes ∆ * 1 and ∆ * 2 of ∆1 and ∆2, and∆ * 3 and ∆ * 3 of∆3 and ∆3 are respectively the convex hulls of vertices (−1, −1, −1), (1, 0, 0), (1,3,9), (1, 3, −1), and (1, 0, −1), resp. (−1, −1, −1), (1, 1, 3), (1,3,9), (1, 3, −1), and (1, 0, −1), resp. (−1, −1, −1), (1, 0, −1), (1, 3, 11), (1, 3, 1), and (1, 0, −2), resp.…”
Section: No 38-no 40mentioning
confidence: 99%
“…Proposition 3.1 and Theorem 3.2 Let ∆ be the reflexive polytope obtained in [7]. For the following transpose-dual pairs, the polar duality extends to a lattice mirror symmetry between the families F ∆ and F ∆ * , where the Picard lattices are given in Table 1.…”
Section: Introductionmentioning
confidence: 99%
“…The strange duality for unimodular singularities is related with the polytope mirror symmetry for families of K3 surfaces that are obtained by compactifying the singularities by Kobayashi [6] in a certain sense. In the study of bimodular singularities, Mase and Ueda [7] extend the duality by Ebeling and Ploog to a polytope mirror symmetry of families of K3 surfaces. More precisely, the following statement is shown :…”
Section: Introductionmentioning
confidence: 99%
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