Landau–Ginzburg models, Calabi–Yau compactifications, and KKP and P=W conjectures

Mirror Symmetry predicts a correspondence between Fano varieties and Landau–Ginzburg models, that is families of varieties (parameterized by complex numbers) with non-degenerate everywhere defined top holomorphic forms. By geometric reasons, as well as because of recent Katzarkov–Kontsevich–Pantev conjectures, P=W conjecture, and the conjecture about the dimension of anticanonical linear system, the challenging problem is to construct log Calabi–Yau compactifications of Landau–Ginzburg models. The canonical class of the compactification has first order poles in components of the fiber over infinity of the compactification. We discuss the notions, constructions, and conjectures mentioned above.