K3 Surfaces

 

The prerequisites are: basic topology (homology and fundamental groups), calculus on manifolds (de Rham algebra and its cohomology), basic Riemann geometry, vector bundles, connections and the curvature. Some understanding of complex algebraic geometry (complex varieties, projective varieties, holomorphic functions, holomorphic line bundles) will be needed as well. General knowledge of Hodge theory (in scope of Griffiths-Harris, chapter 0) will be assumed, but I would repeat all the necessary definitions.

 

 

 

1. Chern classes and the Riemann-Roch theorem for projective surfaces.

2. Topology and geometry of K3 surfaces.

3. Holomorphically symplectic geometry. Hyperkahler structures.

4. Local and global Torelli theorem. All K3 surfaces are diffeomorphic.

5 *. Examples of K3 surfaces. Automorphism group and the shape Kahler cone.

6 *. Nakai-Moishezon theorem. Classification of automorphisms and their dynamics.

7 *. Ergodic action on the Teichmuller space of K3 surfaces. Kobayashi metric and Kobayashi hyperbolic manifolds. Non-hyperbolicity of K3 surfaces.