Math 206 - K3 surfaces

Welcome to Math 206!

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu

Online lectures: WF 1:00-2:20

Course description:

The name "K3 surface" was introduced by Andre Weil in honor of three algebraic geometers, Kummer, Kahler and Kodaira, and the mountain K2 in Kashmir.

This course aims to give an introduction to the geometry of K3 surfaces, focusing on a few selected topics. In particular, I hope to discuss the following:

examples of K3 surfaces
elliptic K3 surfaces
linear systems on K3s
Hodge structures, periods and Torelli
the moduli space of K3s
the tautological ring.

(This may be a bit too ambitious for a one quarter course, so the goals may change slightly as we go.)

Office hours: I am available for questions after lecture or by appointment.

Prerequisites:

I will make an attempt to be as self-contained as the topic permits. However, I will assume working knowledge of algebraic geometry at least at the level of Math 203.

Grading:

The course is intended for graduate students in Mathematics, though everybody is welcome. There will be no homework or exams for this course if you are a math graduate student; the grade will be based entirely on attendance.

Online teaching:

We will use Zoom for both lectures and virtual office hours. Regular attendance is expected (barring unforeseen circumstances and special arrangements). You can access the Zoom Room for this course from Canvas by cliking on "Zoom LTI PRO" at the bottom of the menu on the left. The Zoom Meeting ID is available via Canvas.

The lectures will be given synchronously. Class sessions will be recorded and later made available on Canvas. Sharing the recordings or the links to the recordings with anyone is prohibited.

Class notes: available here and also in Canvas under "Files".

Textbook: None, but you can take a look at Lectures on K3 surfaces by Daniel Huybrechts

Important dates:

First class: Wednesday, January 6.
University Holiday: Monday, January 18.
University Holiday: Monday, February 15.
Last class: Friday, March 12.

Lecture Summaries

Lecture 1: Overview and questions to be addressed in this course. Where K3s stand in the classification of surfaces.

Lecture 2: Preliminaries: curves on surfaces, intersection product, adjunction formula, Riemann-Roch for surfaces. Genus of a K3 surface and connection with genus of curves in the linear series.

Lecture 3: Construction of K3 surfaces via complete intersections in projective space. Genera 3, 4, 5. Construction of K3 surfaces via complete intersections in Fano manifolds of coindex 3.

Lecture 4: Explicit examples of K3 surfaces and of prime Fano manifolds of coindex 3. Genera 6, 7, 8, 9, 10, 12.

Lecture 5: Further discussion of the examples in low genus. Unirationality of the moduli space. Connections between the moduli space of low genus K3s and low genus curves.

Lecture 6: Cyclic covers: formula for irregularity and for the canonical bundle. Construction of genus 2 K3 surfaces as double covers of P^2. Double covers of Hirzebruch surfaces.

Lecture 7: The topology of K3 surfaces. The Betti/Hodge numbers, topological Euler characteristic, the signature of the intersection form, the second integral cohomology as a lattice. Hodge decomposition. The Hodge index theorem.

Lecture 8: The positive cone, nef cone, ample cone. Checking nef and ample via intersections with (-2) curves. Picard-Lefschetz reflections. Action of the Weyl group. Periods and period domain.

Lecture 9: Various realizations of the period domain. Classical symmetric bounded domains. The moduli space of marked K3 surfaces and the Atiyah flop. The moduli space of polarized K3 surfaces.

Lecture 10: The moduli space of polarized K3 surfaces via Hilbert schemes. Very ampleness of 3H if H is ample. Fujita conjecture. Reider's method and strategy for the proof.

Lecture 11: Constructing vector bundles over surfaces via Serre's correspondence. Cayley-Bacharach property.

Lecture 12: Slope stability of torsion free sheaves over surfaces. Simple sheaves. Bogomolov inequality. Proof for K3 surfaces.

Lecture 13: Mumford's proof of Kodaira-Ramanujam. Proof of Reider's theorem. Global generation of big and nef linear series on K3s, and the exceptions. Globally generated linear series over K3s: hyperelliptic and non-hyperlliptic case.

Lecture 14: Projective normality of nonhyperlliptic linear series on K3s. Discussion of hyperlliptic K3s and lattice theoretic characterization.

Lecture 15: Elliptic fibrations and divisors of square zero.

Lecture 16: Elliptic K3 surfaces and Kodaira's classification of singular fibers.

Lecture 17: Comparison between moduli of curves, abelian varieties and K3s. The Hodge bundle. Lambda and kappa classes in each of these settings. Discussion of the tautological ring of M_g. Mumford's relation on M_g.

Lecture 18: The tautological ring of A_g. Mumford relation on A_g, the vanishing of the top lambda class. Poincare duality for the tautological ring of A_g.

Lecture 19: The tautological ring of the moduli of K3s. Expressing kappa classes in terms of the Hodge class. Further results and conjectures.