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.

Lattice duality for families of K3 surfaces associated to transpose duality

“…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.…”

A mirror duality for families of K3 surfaces associated to bimodular singularities

“…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.…”

“…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 :…”

“…Theorem [7] Let ((B, f ), (B ′ , f ′ )) be a dual pair in the sense of [4] of singularities B and B ′ together with their defining polynomials f and f ′ that are respectively compactified into polynomials F and F ′ as in [4]. Then, there exists a reflexive polytope ∆ such that ∆ F ⊂ ∆ and ∆ F ′ ⊂ ∆ * .…”

“…Problem Let ∆ be a reflexive polytope as in [7]. Does there exist general members S ∈ F ∆ and S ′ ∈ F ∆ * such that an isometry T ( S) ≃ Pic( S ′ ) ⊕ U holds ?…”