MAT 626  Topics in Complex Analysis: K3 surfaces and hyper-Kahler manifolds   Fall 2010 

MW 9:10am-10:30am Math 4-130

Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: drop by my office anytime or send an e-mail: kamenova@math.sunysb.edu.

Hyper-Kahler manifolds are (compact) simply connected manifolds with trivial first Chern class which admit an everywhere non-degenerate two-form. In complex dimension 2 hyper-Kahler surfaces are K3 surfaces. Properties and deformations of K3 surfaces are very well studied. Some of the properties can be generalized to higher dimensional hyper-Kahler manifolds.

Syllabus:

1. Generalities on complex compact surfaces.

2. K3 surfaces. Kummer surfaces. Local and global Torelli theorems.

3. Tian-Todorov's theorem on unobstructedness of the moduli space of Calabi-Yau manifolds.

4. Basic results and examples of hyper-Kahler manifolds.

5. Local Torelli theorem. A counter-example of a global Torelli theorem.

6. Finiteness results for hyper-Kahler manifolds.

7. Fibrations and integrable systems. Markushevich's theorem. Matsushita-Hwang theorem.


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