Let M be a compact manifold.
Consider the action of the diffeomorphism group
Diff(M) on the (infinite-dimensional) space Comp(M)
of complex structures. A complex structure is called
ergodic if its Diff(M)-orbit is dense in the connected
component of Comp(M). I will show that on a hyperkaehler
manifold or a compact torus, a complex structure is
ergodic unless its Picard rank is maximal. This result
has many geometric consequences; for instance, it
follows that the Kobayashi pseudometric on any K3
surface or on the deformations of its Hilbert scheme
vanishes, solving a longstanding conjecture by Kobayashi.