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The Geometry of LandauGinzburg Models

 Author / Creator
 Harder, Andrew

In this thesis we address several questions around mirror symmetry for Fano manifolds and CalabiYau varieties. Fano mirror symmetry is a relationship between a Fano manifold X and a pair (Y,w) called a LandauGinzburg model, which consists of a manifold Y and a regular function w on Y. The goal of this thesis is to study of LandauGinzburg models as geometric objects, using toric geometry as a tool, and to understand how K3 surface fibrations on CalabiYau varieties behave under mirror symmetry. These two problems are very much interconnected and we explore the relationship between them. As in the case of CalabiYau varieties, there is a version of Hodge number mirror symmetry for Fano varieties and LandauGinzburg models. We study the Hodge numbers of LandauGinzburg 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 quasiFano hypersurfaces in toric varieties. We describe the structure of a specific class of degenerations of a ddimensional 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 LandauGinzburg model of X and the ddimensional 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 socalled Tyurin degenerations of CalabiYau threefolds to K3 fibrations on their mirror duals and speculate as to the relationship between these K3 surface fibrations and LandauGinzburg models, giving a possible answer to a question of Tyurin. We show that this speculative relationship holds in the case of CalabiYau 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 quasiFano varieties and that the same data describes a K3 surface fibration on its BatyrevBorisov mirror dual. We relate the singular fibers of this fibration to the quasiFano varieties involved in the degeneration of V . We then classify all CalabiYau threefolds which admit fibrations by mirror quartic surfaces and show that their Hodge numbers are dual to the Hodge numbers of CalabiYau threefolds obtained from smoothings of unions of specific blown up Fano threefolds.

 Subjects / Keywords

 Graduation date
 201606

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial 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.