Ergodic complex structures

Let M be a compact complex manifold. The corresponding Teichmuller space Teich is a space of all complex structures on M up to the action of the group of isotopies. The group Γ of connected components of the diffeomorphism group (known as the mapping class group) acts on Teich in a natural way. An ergodic complex structure is the one with a Γ-orbit dense in Teich. Let M be a complex torus or a hyperkahler manifold. I will prove that all Γ-orbits on Teich are dense, except countably many. This result has many applications to complex geometry; I would mention some.