Math 259 - K3 surfaces

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu, AP&M 6-101.

Lectures: WF, 10-11:20am, APM 5218.

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. Also, feel free to drop in if you see me in my office.

Prerequisites:

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

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.

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. General 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. Connection between the moduli space of low genus K3s and low genus curves.
• Lecture 5: Cyclic covers: formula for irregularity and for the canonical bundle. Construction of genus 2 K3 surfaces via cyclic covers. Construction of elliptic K3 surfaces as branched covers of Hirzerbruch surfaces.
• Lecture 6: 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. The Hodge index theorem. The positive cone, nef cone, ample cone. Checking nef and ample via intersections with (-2) curves.
• Lecture 7: Periods and period domain. Torelli theorems stated. Outline of the construction of the moduli space of K3s via periods. The method via the Hilbert scheme.
• Lecture 8: Very ampleness of 3H if H is ample. Fujita conjecture. Reider's method and strategy for the proof. Building rank 2 vector bundles via extensions. Illustration for curves: Max Noether's theorem for canonical embeddings. The Serre construction over surfaces.
• Lecture 9: Slope stability of torsion free sheaves over surfaces. Simple sheaves. Bogomolov inequality. Proof of for K3 surfaces.
• Lecture 10: Crash course in Chern classes. Mumford's proof of Kodaira-Ramanujam. Proof of Reider's theorem. Global generation of big and nef linear series on K3s, and the exceptions for elliptic K3s.
• Lecture 11: Globally generated linear series over K3s: hyperelliptic and non-hyperlliptic case. Projective normality of linear systems over nonhyperelliptic K3s. Lattice polarized K3-surfaces. Description of the elliptic and hyperlliptic loci in terms of lattice polarizations.
• Lecture 12: Characterization of elliptic fibrations in terms of divisors with square zero.
• Lecture 13: Kodaira's classification of singular fibers in an elliptic fibration. Extended Dynkin diagrams.
• Lecture 14: The moduli space of K3 surfaces. Coarse and fine representability. The moduli stack.
• Lecture 15: Crash course on stacks. Strategy for constructing the moduli stack of K3s and what needs to be proved.
• Lecture 16: The construction of the moduli stack of K3s. Smoothness of the moduli stack.
• Lecture 17: Special cycles in the moduli space of K3s. The Picard group. Comparison between moduli of curves, abelian varieties and K3s. The Hodge bundle. Lambda and kappa classes in each of these settings.
• Lecture 18: The tautological rings of M_g, A_g and K_g. Conjectures/results on the largest dimension of a complete subvariety in each of the three cases. Discussion of the tautological ring of M_g. Mumford's relation on M_g.
• Lecture 19: Mumford's relation on A_g and the calculation of the lambda-ring of A_g. The lambda-ring of A_g is Gorenstein. The lambda-ring of K_g and its presentation. The Noether-Lefschetz and kappa rings.