Mirror Constructions for K3 Surfaces from Bimodal Singularities

Given a K3 surface realized as a hypersurface in a weighted projective space or a Gorenstein Fano toric variety, one may construct a mirror K3 surface in various ways. Depending on the precise model, available descriptions of mirror symmetry include the Greene–Plesser mirror, the Berglund–Hübsch transpose construction for invertible polynomials, Dolgachev–Nikulin–Pinkham’s lattice-polarized K3 surfaces, and Batyrev’s reflexive polytope construction. The multitude of descriptions raises the question of whether mirror constructions are consistent. Comparing different mirror constructions often entails making choices—one might need to specify a family containing a K3 surface or a lattice polarization, for example—and thus it is important to establish systematic methods for making these choices.