Modularity of Landau–Ginzburg models

For each smooth Fano threefold, we construct a family of Landau–Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi–Yau varieties with proper superpotential maps; they admit open algebraic torus charts on which the superpotential function $\mathsf{w}$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $\mathsf{w}$ are Dolgachev–Nikulin dual to the anticanonical hypersurfaces in the initial Fano threefold. To do this, we study the deformation theory of Landau–Ginzburg models in arbitrary dimension, specializing to the case of Landau–Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev–Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.