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Regular version of the site

Working paper

Teichmuller spaces, ergodic theory and global Torelli theorem

A Teichm\"uller 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 Γ of M is the group of connected components of the diffeomorphism group. The moduli problems can be understood as statements about the Γ-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. This is my ICM submission, with review of some of my work on Teichmuller spaces and moduli; proofs are sketched, new observations and some open problems added.