Duality of families of K3 surfaces and bimodal singularities

As a generalisation of Arnold's strange duality for unimodal singularities, Ebeling and Takahashi introduced a notion of strange duality for invertible polynomials, which shows a mirror symmetric phenomenon. For each of bimodal singularities, Ebeling produced a Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles by means of its defining equation, which is understood geometrically by Ebeling and Ploog. In my talk, we consider strange-dual pairs of bimodal singularities together with the projectivisations obtained by the one in Ebeling and Ploog's work, by which, we can construct families of K3 surfaces. We discuss whether or not the strange duality extends to dualities of polytopes and lattices for the families. As a consequence, we present that every strange-dual pair can extend to polytope duality, whilst with some exceptions, can extend to lattice duality, and a Hodge-theoretical reason for the lattice duality not being held.Non UBCUnreviewedAuthor affiliation: Tokyo Metropolitan UniversityResearche