We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.
In the last three years a new concept — the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after pa- pers of Seiberg–Witten, Gaiotto–Moore–Neitzke, Vafa–Cecoti and seminal works by Donaldson–Thomas, Joyce–Song, Maulik–Nekrasov–Okounkov–Pandharipande, Douglas, Bridgeland, and Kontsevich–Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin — it is clear that a general “Hodge type” of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications —uniformization, Langlands program to mention a few. In the wall crossing it is also clear that some “Hodge type” of theory exists — Stability Hodge Structure (SHS). This theory needs to be developed in order to reap some mathematical benefits — solve long standing problems in algebraic geometry. In this paper we look at SHS from the perspective of Landau–Ginzburg models and we look at some applications. We consider simple examples and explain some conjectures these examples suggest.