Abstract
Let M be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on M up to the action of the group Diff0(M) of isotopies. The mapping class group Γ:=Diff(M)/Diff0(M) acts on Teich in a natural way. An ergodic complex structure is a complex structure with a Γ-orbit dense in Teich. Let M be a complex torus of complex dimension ≥2 or a hyperkähler manifold with b2>3. We prove that M is ergodic, unless M has maximal Picard rank (there are countably many such M). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic.
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