  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 $\Gamma$ 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 $\Gamma$-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.
