Instructor: Ljudmila Kamenova
Office: Math Tower 3-115
Office hours: drop by my office anytime or send me an e-mail: email@example.com.
Hyperkaehler manifolds are (compact) simply connected manifolds with trivial first Chern class that admit an everywhere non-degenerate 2-form. In complex dimension two hyperkaehler surfaces are K3 surfaces. Properties and deformations of K3 surfaces are very well studied. Some of the properties can be generalized to higher dimensional hyperkaehler manifolds. There have been a number of recent developments in hyperkaehler geometry. This course will focus on several main topics connected with them. K3 surfaces provide intuition for studying higher dimensional hyperkaehler manifolds and we shall start by introducing them. We shall construct Mukai moduli spaces and relate them to Hilbert schemes of points on K3 surfaces. Verbitsky proved a global Torelli theorem for the Teichmueller space of compact hyperkaehler manifolds. We'll discuss vanishing of the Kobayashi metric on hyperkaehler manifolds (this is a recent result joint with S. Lu and M. Verbitsky). We are also going to study twistor spaces and Lagrangian fibrations on hyperkaehler manifolds. If there is interest, we shall discuss automorphisms of hyperkaehler manifolds and connections with dynamics.
1. Generalities on complex compact surfaces.
2. K3 surfaces. Kummer surfaces. Local and global Torelli theorems.
3. Unobstructedness of the moduli space of hyperkaehler manifolds.
4. Basic results and examples of hyperkaehler manifolds.
5. Moduli of sheaves.
6. Twistor spaces and Torelli theorems for hyperkaehler manifolds.
7. Fibrations and integrable systems. Matsushita-Hwang theorem.
8. Finiteness results for hyperkaehler manifolds.
9. Kobayashi's conjectures for K3 surfaces and hyperkaehler manifolds.
10. Automorphisms of hyperkaehler manifolds and connections with dynamics.
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