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Working paper

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 ≥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.