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This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area.

The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.” The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties.

### The Geometry and Moduli of K3 Surfaces

Abstract
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 Kähler cones of K3 surfaces, and give some of their applications.
Andrew Harder, Alan Thompson

### Picard Ranks of K3 Surfaces of BHK Type

Abstract
We give an explicit formula for the Picard ranks of K3 surfaces that have Berglund-Hübsch-Krawitz (BHK) Mirrors over an algebraically closed field, both in characteristic zero and in positive characteristic. These K3 surfaces are those that are certain orbifold quotients of weighted Delsarte surfaces. The proof is an updated classical approach of Shioda using rational maps to relate the transcendental lattice of a Fermat hypersurface of higher degree to that of the K3 surfaces in question. The end result shows that the Picard ranks of a K3 surface of BHK-type and its BHK mirror are intrinsically intertwined. We end with an example of BHK mirror surfaces that, over certain fields, are supersingular.
Tyler L. Kelly

### Reflexive Polytopes and Lattice-Polarized K3 Surfaces

Abstract
We review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.
Ursula Whitcher

### An Introduction to Hodge Structures

Abstract
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kähler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid’s limiting mixed Hodge structure and Steenbrink’s work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.
Sara Angela Filippini, Helge Ruddat, Alan Thompson

### Introduction to Nonabelian Hodge Theory

Flat connections, Higgs bundles and complex variations of Hodge structure
Abstract
Hodge theory bridges the topological, smooth and holomorphic worlds. In the abelian case of the preceding chapter, these are embodied by the Betti, de Rham and Dolbeault cohomology groups, respectively, of a smooth compact Kähler manifold, X.
Alberto García-Raboso, Steven Rayan

### Algebraic and Arithmetic Properties of Period Maps

Abstract
We survey recent developments in Hodge theory which are closely tied to families of CY varieties, including Mumford-Tate groups and boundary components, as well as limits of normal functions and generalized Abel-Jacobi maps. While many of the techniques are representation-theoretic rather than motivic, emphasis is placed throughout on the (known and conjectural) arithmetic properties accruing to geometric variations.
Matt Kerr

### Mirror Symmetry in Physics: The Basics

Abstract
These notes are aimed at mathematicians working on topics related to mirror symmetry, but are unfamiliar with the physical origins of this subject. We explain the physical concepts that enable this surprising duality to exist, using the torus as an illustrative example. Then, we develop the basic foundations of conformal field theory so that we can explain how mirror symmetry was first discovered in that context. Along the way we will uncover a deep connection between conformal field theories with (2,2) supersymmetry and Calabi-Yau manifolds.
Callum Quigley

### Introduction to Gromov–Witten Theory

Abstract
The goal of these notes is to provide an informal introduction to Gromov-Witten theory with an emphasis on its role in counting curves in surfaces. These notes are based on a talk given at the Fields Institute during a week-long conference aimed at introducing graduate students to the subject which took place during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.
Simon C. F. Rose

### Introduction to Donaldson–Thomas and Stable Pair Invariants

Abstract
This chapter is intended for the reader unaccostumed to sheaf counting theories and is meant to serve as a first introduction to Donaldson-Thomas and Stable Pair invariants. We elaborate on some aspects of the expostion in the survey paper by Pandharipande-Thomas. Our emphasis is on one hand on examples that illustrate the properties of the relevant moduli spaces, on the other hand on discussing some of the highlights of the theory.
Michel van Garrel

### Donaldson–Thomas Invariants and Wall-Crossing Formulas

Abstract
We introduce the Donaldson–Thomas invariants and describe the wall–crossing formulas for numerical Donaldson-Thomas invariants.
Yuecheng Zhu

### Enumerative Aspects of the Gross-Siebert Program

Abstract
For the last decade, Mark Gross and Bernd Siebert have worked with a number of collaborators to push forward a program whose aim is an understanding of mirror symmetry. In this chapter, we’ll present certain elements of the “Gross-Siebert” program. We begin by sketching its main themes and goals. Next, we review the basic definitions and results of two main tools of the program, logarithmic and tropical geometry. These tools are then used to give tropical interpretations of certain enumerative invariants. We study in detail the tropical pencil of elliptic curves in a toric del Pezzo surface. We move on to a basic illustration of mirror symmetry, Gross’s tropical construction for $$\mathbb{P}^{2}$$. On the A-model side, we present the proof of Siebert and Nishinou that tropical geometry invariants coincide with classical geometry invariants via toric degenerations. We then summarize Gross’s tropical B-model and the theorem that links the two constructions, emphasizing the common tropical structures underlying both.
Michel van Garrel, D. Peter Overholser, Helge Ruddat

### Introduction to Modular Forms

Abstract
We introduce the notion of modular forms, focusing primarily on the group $$PSL_{2}\mathbb{Z}$$. We further introduce quasi-modular forms, as well as discuss their relation to physics and their applications in a variety of enumerative problems. These notes are based on a lecture given at the Field’s Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.
Simon C. F. Rose

### Lectures on BCOV Holomorphic Anomaly Equations

Abstract
The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky et al. (Nucl Phys B 405:279–304, 1993; Comm Math Phys 165:311–428, 1994). It grew from a series of lectures the authors gave at the Fields Institute in the Thematic Program of Calabi–Yau Varieties in the fall of 2013.
Atsushi Kanazawa, Jie Zhou

### Polynomial Structure of Topological String Partition Functions

Abstract
We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
Jie Zhou

### Introduction to Arithmetic Mirror Symmetry

Abstract
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.
Andrija Peruničić