Families of K3 surfaces and their monodromies (Winter 2020)

          Team taught by Benson Farb and Eduard Looijenga (Tues/Thurs 14:00-15:20 in Eck 202)

K3 surfaces are basic examples of compact complex surfaces.  They are one of the classes in the Enriques-Kodaira classification, and they  are the first nontrivial examples of Calabi-Yau manifolds.  The first part of this class will cover:

1. Basics of K3 surfaces: examples, cohomology, hodge numbers, polarizations, etc.
2. The moduli space of (polarized) K3s via the Torelli theorem.
3. Degenerations of K3 surfaces.
4. Families of K3s and their monodromy representations, and the local monodromy of a degeneration.

A key player in understanding the monodromy of families of polarized K3s is the symplectic mapping class group of a polarized K3 surface. Very little is known about this group.  We will present a possible approach to understanding this group via the Thurstonian viewpoint. In so doing we will go over, among other things:

5. The Thurston classification of diffeomorphisms of (real) surfaces and its relationship to the moduli space of genus g curves.
6. Symplectic mapping class group basics.
7. Geometry of the moduli space of K3 surfaces as a locally symmetric variety.
8. A possible classification of symplectomorphisms arising as monodromies.

Here are some handwritten informal course notes (which we will develop as the  course progresses).