Topics on K3 surfaces (2013 Fall)  

K3 surfaces has been one of the main subjects for the recent algebraic geometry research. It connects to many other topics in algebraic geometry. In this class, we will try to study different aspects of its theory, including linear systems, Hodge theory, moduli space, arithmetic theory etc. One focus will be establishing the global Torelli theorem for K3. If time permits we will also discuss other subjects including the moduli space of its stable sheaves (a large class of examples of hyperkahler variety !); the existence of infinity many rational curves; nonexistence of nontrivial vector fields (in positive characteristic !) etc.   

Prerequisite

Being familiar with basic language on varieties, sheaves, cohomology etc. Basic surface theory is also assumed. A solid knowledge of Hartshorne's book will be a lot more than enough. I also recommend the enthusiastic students taking this class as a tool for helping digesting the materials on Hartshorne. 

Reference: 

[Hu] The main reference is the notes by Daniel Huybrechts. 

[BHPV] For complex K3 surfaces, especially for global Torelli, we also refer to Chapter VIII of Compact Complex Surfaces, second edition, by Barth, Hulek, Peters, Van de Ven. 

[Vo] For more background of Hodge structure, including variation of HS, see Hodge theory and complex algebraic geometry. I, II, by Voisin 

[HL] The geometry of moduli spaces of sheaves. By Huybrechts, Lehn.

Classroom/Schedule

三教 103/Monday 18:40-20:30. 

Exam

The class will have take home exam as the final. The exam sheet is required to be written in tex file!

Syllabus 

9.16: Definitions and examples of K3. See [Hu] Section 1.

9.23: Linear system theory. See [Hu] Section 2.

10.7: Hodge structure: definitions. See [Hu] Section 3.

10.14: VHS and Period mapping of K3. See [Hu] Section 6 and also [Vo] for general background.

10.21: Cones. See [Hu] Section 8.

10.28: Global Torelli I. See [BHPV] Chapter VIII.

11.4: Global Torelli II. See [Hu] Section 7.

11.11: A survey of Moduli space. See [Hu] Section 5.

11.18: Special vector bundles on K3. See [Hu] Section 9.

11.25: NO CLASS.

12.2: Moduli space of sheaves on K3. See [Hu] Section 10. For further reading, see [HL].

12.9: Elliptic K3 (I). See [Hu] Section 11.1-3. 

12.16: Elliptic K3 (II). See [Hu] Section 11.4-5.

12.23: Rational curves on K3 surfaces. See[Hu] Section 13.

12.30: A bird view of supersingular K3. 

To be continued.  

Exercise

I put all homework here. If you want to take the exam, make sure you work on them. 

Final

This is the Final. Please submit your solution by 18:00 Jan. 6 2013. For more instructions, see the exam sheet.  Here is a solution.