Andrew Harder

Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg Models

For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Giventals Landau-Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg models can be expressed as corresponding Laurent polynomials. We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi-Yau varieties.

The Geometry of Landau-Ginzburg Models

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 (C×)d. 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 [141]. 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 ∆, then a bipartite nef partition of ∆ 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.

Families of Lattice Polarized K3 Surfaces with Monodromy

We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.

Calabi–Yau threefolds fibred by mirror quartic K3 surfaces

Abstract We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi–Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi–Yau threefolds, including their Hodge numbers and deformation theory.

Perverse Sheaves of Categories and Non-rationality

In this paper we take a new look at the classical notions of rationality and stable rationality from the perspective of sheaves of categories.

The Geometry and Moduli of K3 Surfaces

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.

Hodge Numbers from Picard-Fuchs Equations

Given a variation of Hodge structure over P1P1 with Hodge numbers (1,1,…,1)(1,1,…,1), we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Moller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zuckers Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.

Introduction to Homological Mirror Symmetry

Mirror symmetry states that to every Calabi-Yau manifold \(X\) with complex structure and symplectic symplectic structure there is another dual manifold \(X^\vee \), so that the properties of \(X\) associated to the complex structure (e.g. periods, bounded derived category of coherent sheaves) reproduce properties of \(X^\vee \) associated to its symplectic structure (e.g. counts of pseudo holomorphic curves and discs).