Teichmuller spaces, ergodic theory and global Torelli theorem

A Teichmuller space Teich is a quotient of the space of all complex structures on a given manifold M by the connected components of the group of diffeomorphisms. The mapping class group G of M is the group of connected components of the diffeomorphism group. The moduli problems can be understood as statements about the G-action on Teich. I will describe the mapping class group and the Teichmuller space for a hyperkahler manifold. It turns out that this action is ergodic. We use the ergodicity to show that a hyperkahler manifold is never Kobayashi hyperbolic.