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. These Landau–Ginzburg models 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; and the general fibres of $\mathsf {w}$ are Dolgachev–Nikulin dual to the anticanonical hypersurfaces in the initial Fano threefold. To construct the family of models, we develop the deformation theory of Landau–Ginzburg models in arbitrary dimension, following the work of Katzarkov, Kontsevich, and Pantev (2017), with special emphasis on the case of Landau–Ginzburg models obtained from Laurent polynomials. Our proof of the Dolgachev–Nikulin mirror symmetry is by detailed case-by-case analysis, which refines Cheltsov and Przyjalkowski's work (published in the same volume of the journal) on the verification of the Katzarkov–Kontsevich–Pantev conjecture.