Ergodic complex structures on hyperkähler manifolds

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 ${\mathrm{Diff}}_0(M)$ of isotopies. The mapping class group $\mathrm{\Gamma }:=\mathrm{Diff}(M)/{\mathrm{Diff}}_0(M)$ acts on Teich in a natural way. An ergodic complex structure is a complex structure with a $\mathrm{\Gamma }$-orbit dense in Teich. Let M be a complex torus of complex dimension $\ge 2$ or a hyperkähler manifold with $b_2>3$. 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.