The Geometry of Landau-Ginzburg Models

• Author / Creator

  Harder, Andrew

• 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.

• Subjects / Keywords

  • Landau-Ginzburg models
  • Toric geometry
  • Fano varieties
  • Calabi-Yau varieties

• Graduation date

  Spring 2016

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R33F4KW4D

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.