Weak Landau–Ginzburg models for Fano varieties

We are going to give a survey of weak Landau–Ginzburg models theory. Due to mirror symmetry conjecture of Hodge structures variations each Fano variety has a dual Landau–Ginzburg model — one-parameterized family of varieties with trivial canonical classes. Periods of such family correspond to come numerical invariants of the initial Fano variety, that is, Gromov–Witten invariants (numbers counting rational curves on the Fano variety). This family is called a weak Landau–Ginzburg model if its total space is a multiplicative torus. In this case the mirror conjecture can be reduced to a quantitative level. We are going to overview some of known weak Landau–Ginzburg models, their conections one to each other, and their connections with some properties of initial Fanos such as Hodge numbers or toric degenerations. A series of open problems are going to be told about.