Modularity of Landau--Ginzburg models

For each 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 potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev--Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau--Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, 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.