K3 surfaces

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

1. Topology and geometry of K3 surfaces.

2. Holomorphically symplectic geometry.
Hyperkahler structures.

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

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

5. Nakai-Moishezon theorem. Classification
of automorphisms and their dynamics.

6. Ergodic action on the Teichmuller space
of K3 surfaces and its applications.
Kobayashi metric and Kobayashi hyperbolic
manifolds. Non-hyperbolicity of 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.

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The time would be Mondays, Wednesdays, 17:00, room 236.