Organizers: Dori Bejleri, Junyan Zhao
Overview
K3 surfaces are among the most fundamental and intriguing objects in algebraic geometry. Defined as smooth, compact, complex surfaces that are simply connected with trivial canonical bundle, they serve as the two-dimensional analogues of elliptic curves and higher-dimensional Calabi–Yau varieties. Their rich structure connects diverse areas such as Hodge theory, moduli theory, arithmetic geometry, and mathematical physics.
The aim of this seminar is to develop a thorough understanding of K3 surfaces from multiple perspectives. We will begin with constructions and examples, including quartic surfaces in projective space, double covers of the plane, and Kummer surfaces. From there, we will explore the cohomology and lattice theory underlying K3 surfaces, highlighting their unique Hodge structures and intersection forms. These tools lead naturally to the Torelli theorems and the study of moduli spaces of polarized K3 surfaces.
Schedules
We meet every Friday from 2 pm to 4 pm with a 10 minutes break.
Introduction and examples of K3 surfaces (Myeong Jae Jeon)
More examples and some linear series (Dori Bejleri)
Stable vector bundles the Bogomolov inequality (Junyan Zhao)
Linear series and Reider's method (Javier Reyes)
Hodge Theory and the Kuga Satake (Myeong Jae Jeon)
The moduli of polarized K3s (Javier Reyes)
The period map and local Torelli theorem (Yu-Chi Hou)
The period map and local Torelli theorem continued (Yu-Chi Hou)
Global Torelli and surjectivity of the period map (Junyan Zhao)
Global Torelli and surjectivity of the period map continued (Junyan Zhao)
Vector bundles and Brill-Noether Theory (Emilio Dominguez)
Elliptically fibered K3 surfaces (Jonathan Rosenberg)
The Rudakov-Shafarevich Theorem and applications (Dori Bejleri)
The ample cone and Kaehler cone (Myeong Jae Jeon)
The ample cone and Kaehler cone continued (Myeong Jae Jeon)
Kulikov models and degeneration of K3 surfaces (Junyan Zhao)
Degeneration of K3 surfaces and more (Junyan Zhao)
References
Huybrechts, Daniel Lectures on K3 surfaces. Cambridge Stud. Adv. Math., 158 Cambridge University Press, Cambridge, 2016. xi+485 pp. ISBN:978-1-107-15304-2