The Geometry of Landau-Ginzburg Models

Abstract

In this thesis we address several questions around mirror symmetry for Fano manifolds and Calabi-Yau varieties. Fano mirror symmetry is a relationship between a Fano manifold X and a pair (Y,w) called a Landau-Ginzburg model, which consists of a manifold Y and a regular function w on Y. The goal of this thesis is to study of Landau-Ginzburg models as geometric objects, using toric geometry as a tool, and to understand how K3 surface fibrations on Calabi-Yau varieties behave under mirror symmetry. These two problems are very much interconnected and we explore the relationship between them. As in the case of Calabi-Yau varieties, there is a version of Hodge number mirror symmetry for Fano varieties and Landau-Ginzburg models. We study the Hodge numbers of Landau-Ginzburg models and prove that Hodge number mirror symmetry holds in a number of cases, including the case of weak Fano toric varieties with terminal singularities and for many quasi-Fano hypersurfaces in toric varieties. We describe the structure of a specific class of degenerations of a d-dimensional Fano complete intersection X in toric varieties to toric varieties. We show that these degenerations are controlled by combinatorial objects called amenable collections, and that the same combinatorial objects produce birational morphisms between the Landau-Ginzburg model of X and the d-dimensional algebraic torus . This proves a special case of a conjecture of Przyjalkowski. We use this to show that if X is “Fano enough”, then we can obtain a degeneration to a toric variety. An auxiliary result developed in the process allows us to find new Fano manifolds in dimension 4 which appear as hypersurfaces in smooth toric Fano varieties. Finally, we relate so-called Tyurin degenerations of Calabi-Yau threefolds to K3 fibrations on their mirror duals and speculate as to the relationship between these K3 surface fibrations and Landau-Ginzburg models, giving a possible answer to a question of Tyurin. We show that this speculative relationship holds in the case of Calabi-Yau threefold hypersurfaces in toric Fano varieties. We show that if V is a hypersurface in a Fano toric variety associated to a polytope delta, then a bipartite nef partition of delta defines a degeneration of V to the normal crossings union of a pair of smooth quasi-Fano varieties and that the same data describes a K3 surface fibration on its Batyrev-Borisov mirror dual. We relate the singular fibers of this fibration to the quasi-Fano varieties involved in the degeneration of V . We then classify all Calabi-Yau threefolds which admit fibrations by mirror quartic surfaces and show that their Hodge numbers are dual to the Hodge numbers of Calabi-Yau threefolds obtained from smoothings of unions of specific blown up Fano threefolds.