Ergodic complex structures on hyperkahler manifolds

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 of complex dimension $\geq 2$ or a hyperkahler manifold with $b_2>3$. We prove that $M$ is ergodic, unless $M$ has maximal Picard rank (there is a countable number of such $M$). This is used to show that all hyperkahler manifolds are Kobayashi non-hyperbolic.

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