The seminar is on Wednesday at 13:30-3:00 pm at Boyd 410.
The goal for the seminar is to understand moduli of K3 surfaces. Our goal to understand a few papers of Alexeev, Brunyate, Engel, and Thompson on KSBA compactifications of moduli spaces of K3 surfaces. (Reference see below). We plan to first learn basic lattice theory and degenerations of K3 surfaces from Scattone's book, and then move to the research papers.
December 10: Kaden Saucedo: Period Maps, Degenerations, and the Boundary of K3 Moduli (Scattone, Chapter 1)
Abstract: This talk introduces the first chapter of Scattone’s thesis on the moduli space of polarized K3 surfaces. We begin by explaining how the period map encodes a K3 surface through its Hodge structure, and how this realizes the moduli space as a quotient of a symmetric domain cut out by the intersection form on the second integral cohomology group. We then describe the basic types of degenerations of families of K3 surfaces and how these are detected by monodromy. The goal of the talk is to give an intuitive picture of how periods, degenerations, and isotropic sublattices fit together, preparing the groundwork for Scattone’s later work that we will look at in the Spring semester.
Janurary 28: Yilong Zhang: Hodge theory of degenerations (Scattone, Section 2.2)
Abstract: This will be the second introductory talk of the learning seminar. First, we will have a quick review of basics on K3 surfaces from Kaden's talk. Then I'll introduce Hodge theory of degenerations, and how the invariants are captured by the lattices. The topic is on Section 2.2 of the book; I will also give a streamlined overview of chapter 3-6.
Feburary 4: Byeol Han: An Introduction to the Baily–Borel Compactifications (Scattone, Section 2.1)
Abstract: In this talk, I will give an introduction to the Baily–Borel compactification of the moduli space of polarized K3 surfaces. Using the period map, this moduli space can be described as a quotient of a Hermitian symmetric domain by an arithmetic group. I will explain what the Baily–Borel compactification is and how it adds boundary points corresponding to degenerations of K3 surfaces. The goal is to give an intuitive understanding of what these boundary components represent geometrically.
Feburary 11: Jaime Ignacio Negrete Gonzalez: Introduction to lattice theory and applications to K3 surfaces (Scattone, Chapter 3)
Abstract: This talk provides an overview of lattice theory and its role in the study of K3 surfaces. We begin with the foundations of integral lattices and discriminant groups, discriminant-quadratic forms and genera of lattices. The discussion then shifts to the geometry of reflections and root systems. Finally, we illustrate how this machinery is applied to K3 lattices, providing the framework for understanding their automorphisms and moduli spaces.
Feburary 18: Kaden Saucedo: Type III Degenerations and Cusps of the K3 Moduli Space (Scattone, Chapter 4)
Abstract: In this talk we discuss the classification of zero-dimensional boundary components in the Baily–Borel compactification of the moduli space of primitively polarized K3 surfaces of a fixed (even) degree, following Chapter 4 of Scattone’s thesis. These boundary points correspond to Type III degenerations of K3 surfaces and can be described in purely lattice-theoretic terms as equivalence classes of primitive isotropic lines in its polarized K3 lattice. We explain how the classification problem reduces to the study of isotropic elements in its discriminant group. We then work through an explicit example to illustrate how the classification can be carried out in practice and how different cusps can be distinguished by the divisibility of isotropic vectors. If time permits, we briefly indicate how these results lead naturally to the study of rank-two isotropic sublattices and boundary curves in the next chapter.
Feburary 25: Yilong Zhang: Torelli Theorem via Type II Degenerations (Friedman 84)
Abstract: In this talk, I'll explain Friedman's proof of Torelli theorem of K3 surfaces using degenerations. We will focus on the degree two case and see how to obtain the Type II boundary components.
March 4: Byeol Han: (Toric geometry/fans)
March 11: Spring break
March 18: Valery Alexeev: (toroidal compactifications)
Future topics: Integral affine structures (Jaime), Kulikov degenerations (Kaden)
Main References:
Scattone, Francesco. On the compactification of moduli spaces for algebraic K3 surfaces. Mem. Amer. Math. Soc. 70 (1987), no. 374, x+86 pp.
Alexeev, Valery; Engel, Philip; Thompson, Alan. Stable pair compactification of moduli of K3 surfaces of degree 2. J. Reine Angew. Math. 799 (2023), 1–56.
Alexeev, Valery; Brunyate, Adrian; Engel, Philip. Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal. Geom. Topol. 26 (2022), no. 8, 3525–3588.
Alexeev, Valery; Engel, Philip, Compact moduli of K3 surfaces, Ann. of Math. (2) 198 (2023)
Various Lecture Notes and supplementary readings:
Lecture notes on K3 surfaces, by Daniel Huybrechts
Notes on compact moduli of K3, by Philip Engel
Modular compactification of moduli of K3 surfaces of degree 2, by Valery Alexeev and Alan Thompson
On lattice-polarized K3 surfaces, by Valery Alexeev and Philip Engel
References for Other Topics:
Torelli of K3 surfaces:
A New Proof of the Global Torelli Theorem for K3 Surfaces, by Friedman
A Torelli theorem for algebraic surfaces of type K3, by Pyatetskii-Shapiro and Shafarevich
Kulikov models and degenerations of K3 surfaces:
Degenerations of K3 surfaces and Enriques surfaces, by Kulikov
Degeneration of Surfaces with Trivial Canonical Bundle, Persson and Pinkham
Type III degenerations of K3 surfaces, by Friedman and Scattone
K3 surfaces with non-symplectic automorphism:
K3 surfaces with non-symplectic automorphisms of prime order, by Michela Artebani, Alessandra Sarti, Shingo Taki
Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution, by Valery Alexeev and Philip Engel
Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism, by Valery Alexeev, Anand Deopurkar, Changho Han
Mirror Symmetry:
Mirror symmetry for lattice polarized K3 surfaces by Dolgachev
Toroidal compactifications of the period domain of K3:
COMPACTICATIONS DEFINED BY ARRANGEMENTS II: LOCALLY SYMMETRIC VARIETIES OF TYPE IV, by Looijenga
Smooth compactification of locally symmetric varieties, by A. Ash, D. Mumford, M. Rapoport, and Y. Tai