Applications of Mirror Symmetry to the Classification of Fano Varieties
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Applications Of Mirror Symmetry To The Classification of Fano Varieties A thesis presented for the degree of Doctor of Philosophy of Imperial College London by Thomas Prince Department of Mathematics Imperial College 180 Queen's Gate, London SW7 2AZ 14th March, 2016 Declaration of Originality I certify that the contents of this dissertation are the product of my own work. Ideas or quotations from the work of others are appropriately acknowledged. Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. Abstract In this dissertation we discuss two new constructions of Fano varieties, each directly inspired by ideas in Mirror Symmetry. The first recasts the Fanosearch programme for surfaces laid out in [1, 21] in terms of a construction related to the SYZ conjecture. In particular we construct Q-Gorenstein smoothings of toric varieties via an application of the Gross{ Siebert algorithm [53,59] to certain affine manifolds. We recover the theory of combinatorial mutation, which plays a central role in [21], from these affine manifolds. Combining this construction and the work of Gross{Hacking{Keel [54,55] on log Calabi{ Yau surfaces we produce a cluster structure on the mirror to a log del Pezzo surface proposed in [1, 21]. We exploit the cluster structure, and the connection to toric degenerations, to prove two classification results for Fano polygons. The cluster variety is equipped with a superpotential defined on each chart by a max- imally mutable Laurent polynomial of [1]. We study an enumerative interpretation of this superpotential in terms of tropical disc counting in the example of the projective plane (with a general boundary divisor). In the second part we develop a new construction of Fano toric complete intersections in higher dimensions. We first consider the problem of finding torus charts on the Hori{ Vafa/Givental model, adapting the approach taken in [96]. We exploit this to identify 527 new families of four-dimensional Fano manifolds. We then develop an inverse algorithm, Laurent Inversion, which decorates a Fano polytope P with additional information used to construct a candidate ambient space for a complete intersection model of the toric variety defined by P . Moving in the linear system defining this complete intersection allows us to construct new models of known Fano manifolds, and also to construct new examples of Fano manifolds from conjectured mirror Laurent polynomials. We use this algorithm to produce families simultaneously realising certain collections of `commuting' mutations, extending the connection between polytope mutation and deforma- tions of toric varieties. Ad Deum qui lætificat juventutem meam Acknowledgements I particularly thank my supervisor T. Coates, A. Corti and A. Kasprzyk, whose instruction in the Fanosearch programme and the surrounding mathematics has been completely invaluable. I would also like to thank T. Coates for the careful reading of this thesis, and the many improvements and corrections he suggested. I would also like to thank my colleagues, particularly M. Akhtar and A. Petracci for a great many useful and enjoyable conversations. I was privileged to learn from several other experts in the field, and particularly thank M. Gross and B. Siebert for their explanations and insightful conversations. I thank the EPSRC for providing the studentship which funded this work. I finally thank my wife, Emily, without whose unwavering support, reassurance and perseverence this thesis would be a mere fantasy. Contents Chapter 1. Introduction 13 1.1. Objectives 13 1.2. The Main Actors: Fano manifolds and Mirror Symmetry 13 1.3. Overview 15 Chapter 2. Fano Manifolds and Mirror Symmetry 19 2.1. Mirror Symmetry for Fano Manifolds 19 2.2. Mutation of Fano polygons 22 2.3. Classifying Orbifold del Pezzo surfaces 26 2.4. The SYZ conjecture and Fano manifolds 27 Chapter 3. Smoothing Toric Fano Surfaces 33 3.1. Introduction 33 3.2. Affine Manifolds With Singularities 33 3.3. From Affine Manifolds to Deformations: an Outline 40 3.4. Log Structures on the Central Fiber 41 3.5. Structures on Affine Manifolds 47 3.6. Consistency and Scattering 53 3.7. Constructing the formal degeneration 60 3.8. Local models at vertices 62 3.9. Smoothing quotient singularities of del Pezzo surfaces 71 3.10. Ilten families 75 3.11. Examples 80 3.12. Conclusion 82 Chapter 4. The Tropical Superpotential 83 4.1. A Tropical Superpotential 83 4.2. A Normal Form for Fano Polygons 85 4.3. Scattering Diagrams from Combinatorial Mutations 88 4.4. Broken Lines in Affine Manifolds 93 4.5. The Affine Manifold B_ 97 P2 4.6. The Proof of Theorem 1.3 100 Chapter 5. The Cluster Algebra Structure 103 11 12 CONTENTS 5.1. Cluster Algebras and Mutations 103 5.2. Background on Cluster Algebras 103 5.3. From Fano Polygons to Seeds 106 5.4. Finite Type Classifcation 109 5.5. Classifying Mutation Classes of Polygons 116 5.6. Relation to the construction of Gross{Hacking{Keel 118 Chapter 6. Complete Intersections and the Givental/Hori{Vafa Model 121 6.1. The Przyjalkowski Method 122 6.2. Finding Four-Dimensional Fano Toric Complete Intersections 124 6.3. Examples 126 6.4. Complete Intersections in Grassmannians of Planes 127 6.5. The Przyjalkowski Method Revisited 131 6.6. Laurent Inversion 135 6.7. A New Four-Dimensional Fano Manifold 139 6.8. From Laurent Inversion to Toric Degenerations 140 6.9. Torus Charts on Landau{Ginzburg Models 142 6.10. From Laurent Inversion to the SYZ conjecture 144 Chapter 7. From Commuting Mutations to Complete Intersections 149 7.1. Overview 149 7.2. Commuting Mutations 150 7.3. Constructing fP;Ξ 151 7.4. The polytope Q~P;Ξ 157 7.5. Recovering toric varieties Xµ(P ) 160 Bibliography 167 CHAPTER 1 Introduction 1.1. Objectives In this dissertation we develop a program which incorporates various techniques from Mirror Symmetry and apply this program to problems related to the classification of Fano manifolds. This program builds on recent advances made by the Fanosearch group (Coates, Corti, Kasprzyk et al.) at Imperial College, which have led to a completely new approach to Fano classification. This perspective, laid out in [21], is based on a formulation of Mirror Symmetry for Fano manifolds explored in [21, 50, 96]. In these papers the authors propose that Mirror Symmetry for Fano manifolds can be understood in terms collections of mirror-dual Laurent polynomials. While a good deal of progress has been made in this area [1,4,22,30,93], there are many important open questions which require a deeper understanding of the situation. We shall particularly focus on two of these questions. • What class of Laurent polynomials is mirror to Fano varieties? • Given a Laurent polynomial f conjecturally mirror to X, how can one construct the variety X? Robust solutions to these would have far reaching consequences. Indeed, given that candi- date mirror-dual Laurent polynomials can be generated relatively easily, these solutions would provide powerful tools to approach the classification of Fano 4-folds. 1.2. The Main Actors: Fano manifolds and Mirror Symmetry Fano varieties are basic building blocks in algebraic geometry, both in the sense of the Minimal Model Program [15,97], and as the source of many explicit constructions. In dimen- 1 sion one there is only one Fano manifold, the Riemann sphere P . In dimension two there are 1 1 2 the famous del Pezzo surfaces: P ×P and the blow up of the projective plane P in 0 ≤ k < 9 general points. In dimension three there are 105 deformation families of Fano manifolds; the classification here was completed by Mori{Mukai in the 1990s, building on work by Fano in the 1930s and Iskovskikh in the 1970s. It is known there are only finitely many deformation families of Fano manifolds in every dimension, but their classification is well beyond the reach of traditional methods. Mirror Symmetry, on the other hand, is a very modern area of mathematics, incorporating many areas of active research. In its original physical formulation Mirror Symmetry proposes 13 14 1. INTRODUCTION the equivalence of the A-twist and B-twist of the superconformal field theory attached to a non- linear sigma model with a K¨ahlermanifold target X with the B-twist and A-twist respectively from a Landau{Ginzburg model with a mirror target X˘. Reconstructing rigorous mathematics from this deep physical phenomenon has produced a wealth of remarkable results. Perhaps most famously was the prediction, via Mirror Symmetry, of the virtual number of rational curves of arbitrary degree on a quintic Calabi{Yau threefold by Candelas{de la Ossa{Green{ Parkes [17], later proved to be correct by Givental [48], as well as in the series of papers of Lian{Liu{Yau [82{85]. Since then, the great industry to formulate the Mirror Symmetry correspondence in a precise mathematical framework has led to the pursuit of many new fields of research. The most dramatic of these is the Homological Mirror Symmetry conjecture of Kontsevich [77] which `lifts' mirror symmetry from a conjecture concerning variations of Hodge structure to a conjecture between A1 categories (a derived category of sheaves on one side, and the Fukaya category on the other). Also prominent is the conjectural formulation of Mirror Symmetry as a duality of special Lagrangian torus fibrations, first made precise in the Strominger{Yau{ Zaslow (SYZ) conjecture [105].