Learning Seminar on K3 surfaces

The seminar is on Wednesday at 13:30-3:00 pm at Boyd 410.

The goal for the seminar is for moduli of K3 surfaces. More precisely, we want to understand a few papers of Valery Alexeev and Philip Engel et al on KSBA compactifications of moduli spaces of K3 surfaces. (Reference see below). Our plan is 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 

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 and degenerations of K3 surfaces

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 Compactification of K3 Surfaces 

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: (Chapter 3)

Feburary 18: Kaden Saucedo: (Chapter 4)

Main References:  

1. Scattone, Francesco. On the compactification of moduli spaces for algebraic K3 surfaces. Mem. Amer. Math. Soc. 70 (1987), no. 374, x+86 pp.

2. 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.

3. 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.

4. Alexeev, Valery; Engel, Philip, Compact moduli of K3 surfaces, Ann. of Math. (2) 198 (2023) 

Various Lecture Notes and supplementary readings:

1. Lecture notes on K3 surfaces, by Daniel Huybrechts 

2. Notes on compact moduli of K3, by Philip Engel

3. Modular compactification of moduli of K3 surfaces of degree 2, by Valery Alexeev and Alan Thompson

4. 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