Weak Landau--Ginzburg models for Fano varieties

Mirror Symmetry studies the correspondence between varieties or one-dimensional families of varieties. Under this correspondence algebraic data of variety reflects symplectic data of its dual symplectic data reflects algebraic data of the dual. The dual to a Fano variety is a Landau--Ginzburg model --- a one-dimensional family of varieties. There is no general method to find a Landau--Ginzburg model for given Fano variety. We observe most of  known approaches for finding them in particular cases. In particular we observe approach of Hori and Vafa for constructing Landau--Ginzburg  models for toric varieties and complete intersections therein, and a method going back to Batyrev for constructing Landau--Ginzburg models  for varieties admitting toric degenerations. All Landau--Ginzburg models  we consider may be interpreted as Laurent polynomials called weak Landau--Ginzburg models.