Dolgachev–Nikulin duality for fibers of toric Landau–Ginzburg models of smooth Fano 3-folds (joint work with A. Harder)

Mirror Symmetry corresponds to Fano varieties certain one-dimensional families which are called Landau–Ginzburg models. Elements of these families are Calabi–Yau varieties mirror dual to anticanonical sections of Fano varieties.
In particular, in the three-dimensional case we deal with Mirror Symmetry of K3 surfaces. One of its most interesting forms is so called Dolgachev–Nikulin duality: it interchanges the lattices of algebraic and transcendental cycles on a K3 surface.
Theory of toric Landau–Ginzburg models provides an effective method of constructing Landau–Ginzburg models of Fano varieties. It is natural to expect that Dolgachev–Nikulin duality holds for fibers of toric Landau–Ginzburg models of smooth Fano threefolds.
This conjecture was proved by Ilten–Lewis–Przyjalkowski in the case of Picard number 1. We establish a certain form of Dolgachev–Nikulin duality for all other smooth Fano threefold, and discuss possible generalisations.