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.
We plan to have one introductory talk in December and then begin the seminar in Spring 2027.
December 10: (10:00-11:30 am at Boyd 628) 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.
Second talk: Byeol Han: time and topic TBA
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 of K3 surfaces:
Degenerations of K3 surfaces and Enriques surfaces, by Kulikov
Degeneration of Surfaces with Trivial Canonical Bundle, Persson and Pinkham
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
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